Okay, so here's the bike we're starting with, roughly -
Mongoose Impasse HDThe fork is a non-tapered, straight 1 1/8" steerer from what I can see, so you'll need a headset with a reducer bearing, which I'm pretty sure Peter sells, but if not, they can be had.
Most of the time when you do a frame swap, the key parts that might not be compatible are everything along the seat tube - bottom bracket, front derailleur, seat post and post clamp.
With a square taper BB, you'll want a frame with a BSA threaded bottom bracket, and as you said, you'll need a new BB as well.
I don't see any mention of the size of the seatpost, you might be able to reuse if it's a 31.6mm, otherwise you might need a new one. Also, no mention of the clamp diameter of the front derailleur, so that's a possible source of needing new parts.
For the rest of the parts, you should probably be able to just swap them over.