If you can get it running as is, you will probably find that BSA folk of the 1930s will be keen to get a look at how many minor details were done, and what the finish was.
This is what largely disappears when bikes are 'restored'.
Early 1930s bikes are particularly uncommon, the depression stifled sales, and bikes of that era were commonly run until completely worn out.
A lot of BSA stuff was common for many years, find out what size and how the piston/rings look and later parts will likely fit.
C&D Autos advertise a lot of parts for older models...