JavaScript (ES9), 227 bytes

This is similar to the Node version below but uses a formula based on parseInt() instead of Buffer() to identify the input verse.

This is ES2018 (aka ES9) because we're using a regular expression with the /s flag (dotAll).

s=>'Mother duck herself1and all23,,Three4two3,Five4four3,Four4three3,One01but none23,Two4one0'.split`,`[parseInt(s,30)&7].replace(/d/g,n=>[x=' little duck',y=/ w.*n/s.exec(s),' of the',x+='s',x+y+'but only '][n])+s.slice(-11)

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How?

In this version, we parse the entire input verse as base 30 (0 to t) and perform a bitwise AND with 7. The parsing stops on the first invalid character, leading to:

 verse | valid part | base 30 -> decimal | AND 7
-------+------------+--------------------+-------
 0 | 'fi' | 468 | 4
 1 | 'fo' | 474 | 2
 2 | 'three' | 23973734 | 6
 3 | 't' | 29 | 5
 4 | 'one' | 22304 | 0
 5 | 'mother' | 554838747 | 3

JavaScript (Node.js),  233 231  227 bytes

Saved 2 bytes thanks to @Shaggy

s=>'Three4two3,Four4three3,Mother duck herself1and all23,One01but none23,,,Two4one0,,Five4four3'.split`,`[Buffer(s)[2]%9].replace(/d/g,n=>[x=' little duck',y=/ w.*n/s.exec(s),' of the',x+='s',x+y+'but only '][n])+s.slice(-11)

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How?

The third character of each input verse can be used as a unique identifier. By taking its ASCII code modulo 9, we get:

 verse | 3rd char. | ASCII code | MOD 9
-------+-----------+------------+-------
 0 | 'v' | 118 | 1
 1 | 'u' | 117 | 0
 2 | 'r' | 114 | 6
 3 | 'o' | 111 | 3
 4 | 'e' | 101 | 2
 5 | 't' | 116 | 8

The output verses are encoded with the following templates:

 verse | template
-------+---------------------------------
 0 | 'Five4four3'
 1 | 'Four4three3'
 2 | 'Three4two3'
 3 | 'Two4one0'
 4 | 'One01but none23'
 5 | 'Mother duck herself1and all23'

Where each digit is replaced with a string according to the following table:

 digit | replaced with
-------+---------------------------------------------------
 0 | ' little duck'
 1 | / w.*n/s.exec(s)
 2 | ' of the'
 3 | ' little ducks'
 4 | ' little ducks' + / w.*n/s.exec(s) + 'but only '

Where the regular expression / w.*n/s extracts this common part from the input:

 went out one day,[LF]
 over the hills and up away.[LF]
 Mother Duck said, "Quack Quack Quack Quack",[LF]

We finally add the last 11 characters of the input, which is " came back.".

Python 3, 267 263 254 bytes

4 bytes saved thanks to @ovs

def f(s):
 for a in zip(T[2:]+T,T):s=s.replace(*a)
 return s
T="8:9:and allHO1BnoneHT2No1T3Nt2F4Nt3FiveINf4MotherD herself"
for r in "H of theI,4ourI,3hreeI,2woI,1neL:,ILs:,L littleD,D duck,NBonly ,Bbut ".split(','):T=T.replace(r[0],r[1:])
T=T.split(':')

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Works by replacing the relevant parts by the respective parts of the next verse.

After the preinitialisation, T is ['8', '9', 'and all of the little ducks', 'One little duck', 'but none of the little ducks', 'Two little ducks', 'but only one little duck', 'Three little ducks', 'but only two little ducks', 'Four little ducks', 'but only three little ducks', 'Five little ducks', 'but only four little ducks', 'Mother duck herself'].

Alternative Python 2, 252 bytes

by @ovs

lambda s:reduce(lambda s,a:s.replace(*a),zip(T[2:]+T,T),s)
T="8:9:and allHO1BnoneHT2No1T3Nt2F4Nt3FiveINf4MotherD herself"
for r in "H of theI,4ourI,3hreeI,2woI,1neL:,ILs:,L littleD,D duck,NBonly ,Bbut ".split(','):T=T.replace(r[0],r[1:])
T=T.split(':')

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QuadR, 257 242 bytes

-14 thanks to Black Owl Kai, -1 thanks to Kevin Cruijssen

ive
Four
hree
Two( little duck)s
One little( duck)
Mother( duck) herself
four
two( little duck)s
only on(e little duck)
but none
and all of the
our
Three
wo
One1
Mother1 herself
Five little1s
three
one1
none of th1s
and all
but only four

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Java 10, 347 bytes

s->String L=" little duck",M="Mother duck herself";int i=9;for(var t:("ive;Four;hree;Two"+L+"s;One"+L+";four;two"+L+"s;only one"+L+";but none;and all of the;"+M).split(";"))s=s.replace(t,++i+"");for(var t:("Five"+L+"s;but only four;and all;none of the"+L+"s;one"+L+";three;"+M+";One"+L+";wo;Three;our").split(";"))s=s.replace(i--+"",t);return s;

Try it online.

Explanation:

We first replace all parts one by one with integers in the range $[10,21]$, and then these integers back to their replacements. The reason this is done in two steps, is because we would otherwise replace replacements.

s-> // Method with String as both parameter and return-type
 String L=" little duck",M="Mother duck herself";
 // Two temp strings to save bytes
 int i=9; // Temp replacement integer, starting at 9
 for(var t:("ive;Four;hree;Two"+L+"s;One"+L+";four;two"+L+"s;only one"+L+";but none;and all of the;"+M).split(";"))
 // Loop over the parts to replace:
 s=s.replace(t, // Replace the part,
 ++i+""); // with the integer pre-incremented by 1
 for(var t:("Five"+L+"s;but only four;and all;none of the"+L+"s;one"+L+";three;"+M+";One"+L+";wo;Three;our").split(";"))
 // Then loop over the parts to replace with in reverse:
 s=s.replace(i--+"", // Replace the (post-decrementing) integer,
 t); // with the replacement-part
 return s; // And then return the modified String as result

T-SQL, 407 390 388 382 bytes

DECLARE @ CHAR(999)=REPLACE(REPLACE('SELECT CASE LEFT(v,2)WHEN''Fi74,''Four''),122,4,''three'')WHEN''Fo74,''Three''),123,5,''two'')WHEN''Th75,''Two''),121,16,''on#'')WHEN''Tw716,''On#''),115,20,''none of th#s'')WHEN''On715,''Mother duck herself''),115,8,''and all'')WHEN''Mo719,''Fiv#s''),113,14,''but only four'')END FROM i',7,'''THEN STUFF(STUFF(v,1,'),'#','e little duck')EXEC(@)

Input is via a pre-existing table $i$ with VARCHAR(MAX) field $v$, per our IO rules.

After a couple of byte-saving REPLACEs, executes the following as dynamic SQL:

SELECT CASE LEFT(v,2)
 WHEN'Fi'THEN STUFF(STUFF(v,1,4,'Four'),122,4,'three')
 WHEN'Fo'THEN STUFF(STUFF(v,1,4,'Three'),123,5,'two')
 WHEN'Th'THEN STUFF(STUFF(v,1,5,'Two'),121,16,'one little duck')
 WHEN'Tw'THEN STUFF(STUFF(v,1,16,'One little duck'),115,20,'none of the little ducks')
 WHEN'On'THEN STUFF(STUFF(v,1,15,'Mother duck herself'),115,8,'and all')
 WHEN'Mo'THEN STUFF(STUFF(v,1,19,'Five little ducks'),113,14,'but only four')END
FROM i

Uses a CASE statement and STUFF commands to insert/overwrite characters at the listed positions.

EDITS:

1. Replace original (below) with an entirely different strategy


2. Saved two bytes by switching to LEFT instead of SUBSTRING and eliminating a space


3. Saved 6 bytes by changing variable to CHAR and moving an extra letter into the second REPLACE (thanks, @CDC!)

Here's my first version, using a different method (post-replacement, formatted):

DECLARE @ VARCHAR(MAX)
SELECT @=v FROM i
SELECT @=REPLACE(@,PARSENAME(value,2),PARSENAME(value,1))
FROM string_split('e.but none.and all
 -e.One little duck.Mother duck herself
 -o.only one little duck.none of the little ducks
 -o.Two little ducks.One little duck
 -r.two little ducks.one little duck
 -r.Three.Two
 -u.three.two
 -u.Four.Three
 -v.four.three
 -v.Five.Four
 -t.and all of the.but only four
 -t.Mother duck herself.Five little ducks','-')
WHERE LEFT(value,1)=SUBSTRING(@,3,1)
PRINT @

STRING_SPLIT and PARSENAME are used to break a string into rows and columns via - and . separators.

The first column is a key character that is matched against the 3rd letter of the input verse (thanks for the idea, @Night2). The second and third are the replacements being performed for that verse.

Python 2, 1034 bytes

This is my code! It employs a simple dictionary. After running this code, you can enter any verse and it will output the next verse.

PS: I am new to this channel and this is my first post. I really liked this challenge so decided to give it a try. Please feel free to correct me.

import sys
i=sys.stdin.readlines()
s="Mother":"""Five little ducks went out one day,
over the hills and up away.
Mother Duck said, "Quack Quack Quack Quack",
but only four little ducks came back.""",
"Five":"""Four little ducks went out one day,
over the hills and up away.
Mother Duck said, "Quack Quack Quack Quack",
but only three little ducks came back.""",
"Four":"""Three little ducks went out one day,
over the hills and up away.
Mother Duck said, "Quack Quack Quack Quack",
but only two little ducks came back.""",
"Three":"""Two little ducks went out one day,
over the hills and up away.
Mother Duck said, "Quack Quack Quack Quack",
but only one little duck came back.""",
"Two":"""One little duck went out one day,
over the hills and up away.
Mother Duck said, "Quack Quack Quack Quack",
but none of the little ducks came back.""",
"One":"""Mother duck herself went out one day,
over the hills and up away.
Mother Duck said, "Quack Quack Quack Quack",
and all of the little ducks came back."""
print s[i[0].split(" ")[0]]

PHP (7.4), 253 247 bytes

-6 bytes by improving how the replacements array is built with more help from "Unpacking inside arrays".

<?=strtr($argv[1],array_combine([0,1,...$a=[Five.$l=($o=" little duck").s,($b="but only ").four.$l,Four.$l,$b.three.$l,Three.$l,$b.two.$l,Two.$l,$b.one.$o,One.$o,"but none of the$l","Mother duck herself","and all of the$l"]],[...$a,$a[0],$a[1]]));

Try it online!

This creates an array of every possible replacement (12 used + 2 unused) in a key=>value format. Example: ['Mother duck herself' => 'Five little ducks', etc...] and then just replaces those using strtr.

The only interesting thing is my first ever usage of "Unpacking inside arrays" which is a new feature in PHP 7.4.

PHP, 264 bytes

<?=str_replace(($a=[[Five.$l=($o=" little duck").s,($b="but only ").four.$l],[Four.$l,$b.three.$l],[Three.$l,$b.two.$l],[Two.$l,$b.one.$o],[One.$o,"but none of the$l"],["Mother duck herself","and all of the$l"]])[$i=strpos(vuroet,($v=$argv[1])[2])],$a[++$i%6],$v);

Try it online!

I have stored different words of each verse in an array. I find which verse the input is using third character of the input as it is unique (vuroet). Then I simply replace different words of that verse with different words of the next verse.

Clean, 352 bytes

import StdEnv,Text
$n=#"Five"+e+"four"+d,"Four"+e+"three"+d,"Three"+e+"two"+d,"Two"+e+"one"+a,"One"+a+b+"but none"+f,"Mother duck herself"+b+"and all"+f.[indexOfn.[2]"tvuroe"]+" came back."
a=" little duck"
b=" went out one day,nover the hills and up away.nMother Duck said, "Quack Quack Quack Quack",n"
d=a+"s"
e=d+b+"but only "
f=" of the"+d

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C# (Visual C# Interactive Compiler), 262 bytes

x=>$@"l[g=x[2]*37%724%7]g-5:; little duckg-4: herself;s; went out one day,
(x=x.Split('n'))[1]
x[2]
g-5:and;but g-4:all;only;none (g>3?"of the":l[g+1].ToLower()) little ducg%3:ks;;k came back.";var l=" Four Three Two One Mother".Split();int g

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PowerShell, 356 343 340 336 bytes

param($p)$l,$M=' little duck','Mother duck herself'
("ive!our;four!three;hree!wo;Four!Three;two!one;s c! c;Two!One;s w! w;only one!none of the;k c!ks c;One$l!$M;but none!and all;$M!Five$l`s;and all of the!but only four"-split';')[$(switch -r($p)^Fi0,1^Fo2,3^Th2,4,5^Tw6..9^O10,11^M12,13)]|%$p=$p-creplace($_-split'!')
$p

Try it online.

More readable version:

param($p)
$l, $M = ' little duck', 'Mother duck herself'
$replacements = @"
ive!our
four!three
hree!wo
Four!Three
two!one
s c! c
Two!One
s w! w
only one!none of the
k c!ks c
One$l!$M
but none!and all
$M!Five$l`s
and all of the!but only four
"@ -split 'n'
$i = switch -regex ($p) ^Fi 0, 1 
 ^Fo 2, 3 
 ^Th 2, 4, 5 
 ^Tw 6..9 
 ^O 10, 11 
 ^M 12, 13 
$replacements[$i] | % $p = $p -creplace ( $_ -split '!' ) 
$p

PowerShell, 265 263 255 251 246 bytes

$d='Five1four2s,Four1three2s,Three1two2s,Two1one2,One23but none52s,Mother duck herself3and all52s'-split','
'2s34',' little duck',-join($args-split'(s)')[5..39],'but only ',' of the'|%$d=$d-replace++$i,$_
$d[+"$args"[2]*37%724%7]+' came back.'

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I used the brute force to find the expression +"$args"[2]*37%724%7.

 verse | 3rd char. | ASCII code | *37%724%7
-------+-----------+------------+-----------
 0 | 'v' | 118 | 1
 1 | 'u' | 117 | 2
 2 | 'r' | 114 | 3
 3 | 'o' | 111 | 4
 4 | 'e' | 101 | 5
 5 | 't' | 116 | 0

Thanks @Arnauld for the 3rd char.

Japt v2.0a0, 143 bytes

Tried encoding a single verse with replacements but, in the end, adapting Arnauld's solution ended up being shorter. Have another idea that might, hopefully, work out shorter again but don't know when I'll get to try it.

tBn)i`Fr4È(e3
Two4e0
MÇ@r Ýõ Êelf1d a¥23
O01¿t Í
23
TËG4two3
Five4fr3`·g`v`b¢Î)rdÈ°g[V=` Ò¤ Ýõ`W=Uf/ w.*n/s ` e`V±'sV+W+`¿t § `]

Try it - includes all verses

Bash, 373 355 bytes

Nothing too crazy here. An easy few byte reduction would be to replace the two-character variables (a1,a2,a3,e1..e6) with single character ones.

read a1..3 b
read c
read d
read e1..6
W="$e1 $e2"
X="$e3^ $a2 $a3"
Y="$e5 $e6"
Z="$e4 $Y"
p=$X
case $a1 in M*)p="Five $Y::12";s="but only four $Y";;O*)p="$d::11 herself";s="and all of $Z";;Tw*)p=$X/s;s="$e1 none of the $a2 ducks $e6";;Th*)s="$W one $e4 duck $e6";;Fo*)s="$W two $Z";;Fi*)s="$W three $Z";;esac
echo $p $b;echo $c;echo $d;echo $s

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Try it online!

05AB1E, 134 bytes

“€µ‚•„í†ìˆÈ“#’ „ê Ðœs’δJ樅î¥Ðœº¶s‚ìðδJD…€³€É δ쨦“€ƒ€Ÿ€‚€€““€³Šª€‚€€“‚’ „ê Ðœs ’δJ셋邃.δJU.•4Ôāl•|н2èk©è.ª?I„ 
ý#3.$17£ðý„ 
¶:,X®è?

Try it online!

Since i'm relatively new to 05AB1E, this can maybe be golfed a lot

 ### Preparation of the output strings ###
“€µ‚•„í†ìˆÈ“ push "one two three four five"
# split that by spaces
’ „ê Ðœs’ push " little ducks"
δJ join each number with " little ducks"
ć¨ separate "one little ducks" and drop the "s"
…î¥Ðœº¶ push "mother duck herself"
s swap it with "one little duck"
‚ì prepend both strings to the list ["mother duck herself", "one little duck", "two little ducks" ... ]
ðδJ append a space to each list entry
D duplicate it
…€³€É push "but only "
δì prepend "but only " to each list entry
¨¦ drop the first and last list entry
“€ƒ€Ÿ€‚€€“ push "and all of the"
“€³Šª€‚€€“ push "but none of the"
‚ push the two strings into a list
’ „ê Ðœs ’δJ append " little ducks " to each
ì prepend it to the sentence list ["and all of the little ducks ", "but none of the little ducks ", "but only one little duck " ...]
…‹é‚ƒ. push "came back."
δJ append that to each list entry
U save that list in X for later use

 ### Determine which verse has to be answered ###
.•4Ôāl• push "eoruvt"
|н2è get the third letter of the input
k get the index of that letter in "eoruvt". Now we know which verse we must return
© save that index in ® for later use

 ### Print the answer strings ###
è.ª? print that index of the first sentence list (first three words of answer)
I„ join the four input strings by <space><newline>
ý
# split that by spaces
3.$ cut off the first three words
17£ keep only the next 17 words
ðý join remaining words by spaces
„ 
¶: replace <space><newline> by only <newline>
, print that ("went out ... Quack",") 
X®è? print the last line of answer

Perl 6, 247 bytes

S:i/nesl.*?<(s//om/..(.)(S*)**3%s(.**92).*(.**11)/;[(my$l=" little ducks")~$3 XR~"Mother duck herself$2and all of the","One$l~=$2but none of the","Five$l~="but only "four","Four$lthree","Three$ltwo","Two$lone"]["eotvur".index($0)]

Try it online!

Definitely golfable, especially the last 5 elements in the list in the form "num $l num-1", or the starting regex that matches the right parts of the old input.

Charcoal, 156 bytes

≔⌕tvuroe§θ²δ§⪪”↶0∨↘»≔xj➙⌈´βＸPＮLA‽⟦O⧴＆▷V'¦³≧ψZρ⊞t”¶δＦ‹δ⁵”↶↧V4ⅉＨ‴G%”Ｆ‹δ⁴s⮌…⮌θ¹⁸⸿η⸿ζ⸿§⪪”}∧h⁴ＮＱ≕Q^⪫ΦＧ✂q'ⅉＭG./_⸿s⁵6P⁴″⊟±ＮNpOfBz↷Ｆι‖ＴM→⁻γ？k⁴ς!d⁵º'E,θ}x§-υ”¶δ✂ε±¹¹

Try it online! Link is to verbose version of code. Explanation:

≔⌕tvuroe§θ²δ

Look at the third character of the first line to work out which verse we want.

§⪪”↶0∨↘»≔xj➙⌈´βＸPＮLA‽⟦O⧴＆▷V'¦³≧ψZρ⊞t”¶δＦ‹δ⁵”↶↧V4ⅉＨ‴G%”Ｆ‹δ⁴s⮌…⮌θ¹⁸

Output the first part of the first line by indexing into the list of strings Five, Four, Three, Two, One, Mother duck herself. Then print little duck and s as appropriate, followed by the last 18 characters of the input line (which are always the same on each verse).

⸿η⸿ζ⸿

The middle two lines are always the same on each verse.

§⪪”}∧h⁴ＮＱ≕Q^⪫ΦＧ✂q'ⅉＭG./_⸿s⁵6P⁴″⊟±ＮNpOfBz↷Ｆι‖ＴM→⁻γ？k⁴ς!d⁵º'E,θ}x§-υ”¶δ✂ε±¹¹

For the last line it turns out to be golfier to include little ducks in the list of alternatives for some reason, but the last 11 characters are still copied from the input.

ink, 353 bytes

=d(v)
~temp n=(v?"Fi")+(v?"F")*4+(v?"Th")+(v?"T")*2+(v?"O")
n-1:n:n-5:n-4:n-3:One little duckn-2:s went out one day,
over the hills and up away,
Mother Duck said "Quack Quack Quack Quack",
and all of the little duckn-3:s came back.

Try it online!

First, uses substring checks to figure out what verse we're in - that's easy enough thanks to the capitalisation of the numbers at the start - F, T and O don't occur in any other places, and you can distinguish the second and fourth verse from the first and third by also checking for Fi and Th respectively.

Then we just do what ink does best and print plain text specked with conditionals. I tried using switch statements at first, but while that looked nicer, it actually ended up longer.

You'd think the Quacks would be a good place to use variables, what with a string being repeated a bunch, but variables come with enough overhead that every way I tried to do that made the code longer. Maybe that's a sign I'm not supposed to golf in ink.