Here's a couple pics. The first one, engine and frame numbers. I know you said you don't have
any engine numbers. Look
closely at your surface where the numbers should be. I'm talking good lighting and a magnifying glass. Does the surface look exactly like the cast surface elsewhere on the cases? BSA sometimes put periods or dots in the numbers and these got pretty deep. Is there any sign of those left? The original numbers for a '68 Firebird Scrambler would start with A65FB, the B indicating 1968. Those numbers would be on both the frame and the engine.
It's possible (but not likely) that you have a set of replacement cases. Dealers who rebuilt engines with replacement cases were supposed to stamp the numbers back into them, but they didn't always. (Same with replacement frames, and it's not as uncommon to find an OIF frame unstamped. A lot of those were replaced under warranty.)
The frame numbers in the first pic, a late '66 Lightning, were just stamped into the frame lug at the front motor mount. (BSA began using matching numbers in early calendar 1966, which was about the middle of the production year.) Later on, the frame lug was modified to have a flat pad. (see the second pic, a '69 Lightning with the numbering system BSA started using in the middle of that production year. The '69 and newer BSA twins also had the engine number on a raised pad, with little BSA logos lightly etched in the surface. Much harder to fake.) The third pic shows a frame number from '65. The '62-'65 unit twins had the frame numbers up closer to the neck and they all started with A50 regardless if the engine was an A50 or an A65. Those engine and frame numbers did not match. OIF frames (not shown) had the numbers back up by the neck too.
If your frame has no numbers, either it's an aftermarket frame, (pics would help, you must know someone with a camera.) or it's a stock frame with the numbers removed. Again, look really close at the areas where the numbers should be. If you see any evidence that the numbers, engine or frame, have been removed, I would take it back where I got it, explain to the guy he can buy it back or explain to the police why he sold you a motorcycle with altered numbers.