Well its called a Oversight.... these happen and well with Ridley Scott we have seen a few over the years...
Ultimately we dont know what happened 100% As it happens off screen.. but there are only TWO options.
1) David simply Changes Clothes, and then has enough time to do this, and add the Scars, and take the Hand Off. The Problems with this... are the hole in his chin, i dont think i remember it being there latter, and the voice box went a bit funny after the fight but it was only minor and so could have fixed itself.
2) David Uploads his Mind into Walters Body, again how would he do this? How long would it take to transfer all those Emotions. The Problem with this is this Walter does not remember Daniels mentioning a Cabin on the Lake.... unless when he Transfers his Mind/Memories over he Erases everything of Walters and so he has no knowledge of any Memories Walter has.