Unfortunately (I guess) the simplest, easiest to replicate cartridge-firing handgun would probably be an open-bolt automatic. Of course, even a semi-automatic open bolt gun is considered a machine gun by the BATFE, but it would technically be the simplest as it only needs a receiver tube, barrel, magazine, bolt (with firing "pin" nub machined into the breech face), recoil spring, and a trigger mechanism basic enough to hold the bolt back when released but to simply drop out of the way when pulled.
So, something like an M3 "grease gun" or a Sten.