So after playing Final Moment, you can eliminate opponents characters and win... but what happens if you eliminate their hand and deck?

As I read it “after the action phase ends”, is before upkeep. Is there any way to read after upkeep as still fulfilling after action requirement? I’ve got a local that wants to use it with a mill deck and I’m not seeing a way for it to work

I think you're right.

The action phase ends, at this point Final Moment triggers, then play proceeds to the upkeep phase, the "mill" win condition specifies "at the end of a round (after the upkeep phase)"

So yeah, I would suggest you cannot play Final Moment and win by mill.

I would think you still lose. After the actions phase ends you lose the game from Final Moment. This happens before the next upkeep phase where the opponent would lose due to no cards in hand and not able to draw a card.

Final Moment, best card ever when you need to run to the bathroom.