According to this link http://hadland.files.wordpress.com/2012/07/bsa.pdf BSA produced a special hub in 1954 that allowed the rear wheel to be removed without disturbing the chaincase. The tech sheet says that the operating chain was located on the left hand side which clearly yours is not. So now we can presume a build between 1943 - 1954. If the reflector has a date cast in to it that falls between those years, odds are good that is the build year.

The taillight has what appears to be a 65 or 66 cast in to it and so probably was a later addition.