Determining the exact date and model of a BSA motorcycle is not an exact science.
We rely on information obtained from a number of sources, most notably the
as well as the collected experience of a number of BSA enthusiasts. Even then, there are subtleties which require an expert's experience.
Even if it were simple, it is now almost 30 years since BSA went out of business, so there is a strong chance that the engine and frame did NOT leave the factory together. The definitive list is at : UK BSA Owners' Club - BSA Machine Numbering System
Enter your engine number and frame number in the What BSA is that online form.