I was unclear from the 5e PHB if the time needed to copy a spell into my spellbook needed to be consecutive (like my character sat down and just dug in), or if he could practice a little here and a little there to get it done.

Does the time required to copy a spell into a spellbook have to be consecutive, or is that just the cumulative time required?

Most likely cumulative time - but DM may decide otherwise

The rules on copying spells is listed in the PHB(emphasis mine):

Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.

It doesn't say that is must be concurrent or consecutive, so the implication is that it's just a total time that needs to be spent and can be cumulative.

DM's alternative

However, it's still loose enough that a DM could reasonably rule that they want it to happen all at once:

Focusing on something for two hours is different than spending 10 minutes/day on something over 12 days. A DM could rule that they require this and because the language doesn't state one way or the other definitively, still be 'correct' beyond rule zero of DM's prerogative.

This breaks down for higher level spells

The problem with forcing consecutive is when you get to higher level spells. A 9th level spell will take 18 hours to copy. That's an entire day spent doing nothing but copying. While that is possible, it's more probable that the time required can be split it up.

I'd argue Consecutive

I can't find a rule determining this, so I'm going to go off of what consensus in the books and my own opinion.

Most things that have a duration are meant to be consecutive. Things like long rests or casting spells have specific situations where they can be interrupted and must be tried again.

So personally I'd rule and have ruled that scribing spells has to be consecutive. Not to mention that tracking 2-18 hours "here and there" is can be annoying and will significantly lenghten the time passed before you have a new spell.

At my table

We've mainly used this ruling, simply because we don't want to track the 2 or more hours or say In 3 days you learn X. This doesn't really give us a problem since we can always find these 2 hours. Traveling on a cart, just before rest, or after. It hasn't happened often that we were so desperate to learn a spell that we needed to track it like that, just so it'd be ready for the BBEG. This may be an entirely different matter if you have high stakes based on a ticking clock.
Spells higher than first level are usually scribed during downtime or during travel. For our campaign it is quite often that we travel longer than a few days, which gives us plenty of time.

Even for a 9th level spell this'll be 18 hours, which is slightly less than a day (or 1 day and 10 hours with a long rest before and after). But again, this is mostly done during downtime.

For crafting items we use a completely different system so I can't comment on that.