I guess it depends how well damped the stud wall, as a membrane, is. If it’s flexible (unbraced to the outside wall) but there’s not enough loss to damp it critically (or close to) then it will act as a soundboard, like a piano’s, and influence the sound. If it’s as stiff as concrete then it will act as a reflector, although to be honest it will probably be a pretty good reflector independent of what Paul does to it.
I wouldn’t worry about sound to/from the outside wall. It’s not as if the Westway runs anywhere near, so the amount of sound coming in via all that stonework shouldn’t be large, and as for sound going out, well, how near are the neighbours exactly ?