The Military Roadsters of the war years were adapted from very similar civilian models. Chromium was expensive in peace time, and practically unobtainable during the war years, so many of the larger manufacturers offered an "all black" option in the thirties, and most civilian bikes produced during the war years came with black-painted fittings. I think it would be quite appropriate to restore your bike as an all-black model.
The top-tube gear lever would be a Sturmey-Archer "quadrant" shifter. Handlebar-mounted shifters didn't become common on roadster models until after the war. You could rebuild the bike as a singlespeed initially while you hunt for the appropriate 3-speed components. This stuff pops up on eBay all the time, and VCC cycle-jumbles can be a good source of old roadster parts.
I'm sure you've found the "BSA & Military Bicycle Museum" site already, but just in case, here it is:
along with its partner site, the "Online Vintage Bicyle Museum":
it's an excellent source of pictures and original documentation.
Calmes dans le demi-jour
Que les branches hautes font,
Pénétrons bien notre amour
De ce silence profond.