We can calculate the C number with the frame number of the BSA. You mention a plate number is that actually the frame number? If it is your tank number would be C4197173
I have some more information about this BSA if you are interested but I need the frame number to find out. Best send me an email.
My calculations agree with Rob's serial of C4197199. It would originally have had the large fuel tank and valanced rear mudguard but by this time, MT110 rear lamp rather than the early 1940 season MT1130 and the speedo drive was of the later type.
On the end of this row is C4197202 in service with New Zealand forces in the UK during 1940. Three different is as close as I can get.
While I'm restoring my bike, I guess I have to put the C number on the mudguard board as like on the picture above? Nowadays when i look on the internet, I see similar bikes where the C number has been written down on the tank instead of the mudguard board. They have sprayed the mudguard board totally green. Is there a reason for?