Can you perform a Ready action with the condition being that you complete your bonus action, then do your bonus action, and then complete your readied action?

For instance, say I have the Shield Master feat. I Ready the Attack action with the condition that the enemy is prone, then I take the bonus-action shove from Shield Master and knock him prone before I take my readied Attack action.

Is this a legal exploit to reverse the action order?

No, your Shield Master example won't work

Shield Master says:

If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feet of you with your shield.

If you take the Ready Action you are not taking the Attack action, even if you use it to attack.

you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn.

Only once the trigger occurs and your attack completes are you counted as having taken the Attack action.

Since you don't take the Attack action, you do not get a bonus action shove. Since you don't get a bonus action shove, you won't be able to trigger your Readied action.

So in the end, attempting this only results in a wasted action. It is still a valid way to phrase the Ready action, it will just always result in no trigger happening (unless the enemy becomes prone and right next to you still for some other reason before your next turn).

In general, you can use the results of a bonus action as a trigger for a Ready action

Though the above example didn't work, there is no reason that you cannot use the results of a bonus action to trigger a Ready action. For example, you can have say after "I cast this bonus action spell, do X" or "after I take my bonus action attack, do X". As long as you are able to do the bonus action in the first place (unlike with the Shield Master bonus action) and you can perceive it, then you can use it as a trigger.

Note that you can't just set a trigger off of "when I use my bonus action" because bonus actions are not a thing in-universe and thus it is not perceivable. The results or actual actions done with the bonus action are of course valid as described above.

You are asking two different questions here.

Your first question is:

Can you perform a ready action with the condition being that you complete your bonus action, then you do your bonus action and then complete your action from your ready action?

And the answer is that yes, you can do that. However, you don't really get any benefit from that, since any bonus action you could take to trigger the action must be a bonus action you could take anyway. You're just as well off not readying an action, using your bonus action, and then just taking a normal action.

Your second question is:

Is this legal exploit to reverse the action order?

The Shield Master feat says:

If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feet of you with your shield.

Though you could ready an action to trigger off of the "try to shove... with your shield" bonus action, you would never actually get to use that reaction and will have effectively wasted your action.

The important text is "If you take the Attack action on your turn...". The Attack action is distinct from an attack and from a regular action. If you've readied an action, even if your readied action is to take the Attack action, then you still haven't taken the Attack action. Since you don't actually take the Attack action, your bonus action of "shove... with your shield" never takes place during your turn.

In this scenario, you do still get to attack if, after readying that action, an opponent somehow ends up both adjacent and prone to you. However, you will still lose the shield shove bonus action if this happens outside your turn.

While your particular example won't work due to Shield Master's bash requiring the Attack action be taken first, you can indeed have a readied action trigger on your own turn off of an (otherwise valid) bonus action. Your reaction is available from the beginning of your turn, meaning you can trigger it before your turn is over.

The only limitation on readied action triggers is that they must be "perceivable circumstance[s]", there's nothing in that barring your bonus action from being the trigger.

It Could Destroy the Universe

This is a little like the time travel paradox of what would happen if a person traveling back in time caused themselves to never have traveled back in time in the first place.

Even if the Attack action were not distinct from a Ready action resulting in taking the Attack action, you would have a conundrum. The bonus actions made available by taking the attack action require it actually be taken. You would never be able to use the bonus action in the first place if the circumstances for the readied action fail to trigger.

In your example, failing to knock the enemy prone would result in never triggering the readied action, never getting the bonus action shove, and thus never being able to try to knock them prone to begin with. Naturally this would destroy the space-time continuum of your campaign world.

And yes, that's an absurd way of putting it, but the mechanical issue stands none the less. Things can't be used to trigger their own prerequisites unless success is absolutely certain.

It may well be that the logic of a bonus action not being able to come before an attack it depends on is based in a less convoluted version of its own logic. It is always possible that you will not actually take the prerequisite attack. If this is the case then this proposed exploit is just a roundabout way to get to the exact same original problem.