This came up today in a game of Commander today. Player A has a Wood Elves and The Mimeoplasm in his graveyard. Player B has a Body Double in his. Player A casts Exhume.

Player A wishes to bring back The Mimeoplasm exiling Body Double and Wood Elves. Player B wishes to bring back Body Double copying The Mimeoplasm and then exiling The Mimeoplasm and the Wood Elves.

So many replacement effects… I believed that the first The Mimeoplasm exiling would happen because APNAP, and since Body Double was no longer coming into play, none of its effects would take place anymore, as there's nothing to replace. Is this how it works, or am I missing something?

This is essentially a simpler, less-crazy-to-set-up variation on this scenario, and the same answer applies here as there: since Body Double is already changing zones at the same time as The Mimeoplasm, it cannot also be chosen to change zones for Mimeoplasm's ability.

Assuming there's nothing else in anyone's graveyard, Mimeoplasm isn't going to be able to exile two creature cards, so it won't be able to copy anything. Body Double will be able to copy The Mimeoplasm, since it's in the proper zone at the time the replacement effect is applied and Body Double isn't trying to get The Mimeoplasm to change zones. But that isn't really all that great an idea, since the Double-plasm won't be able to copy anything either.

If there are additional creature cards in graveyards, Player A will get first choice of what they want to exile, and if Player B copies The Mimeoplasm they'll have to exile something else, since the cards Player A chose will be off-limits for exiling. (Though if Player B wishes to Body Double something other than The Mimeoplasm instead, the cards chosen by Player A will be a legal choice for that.)