Rather casually I suspect. In the early 70s - OK when BSA were already in trouble - I took an A65 engine to the Small Heath Service Dept for repair. You just parked up outside the factory doors and walked in and met someone in a workshop coat. They made some notes on a pad and took the engine away. I seem to recall there was a metal work bench that you put your engine on and the workshop was behind some swing doors. We were told to come back in 3 days which we did and the engine was ready to pick up. I think it had a new bottom end and the bill was £36. It was all had written on a Kalamazoo pad I appreciate this was not a replacement engine but the whole process was very "old fashioned" and not at all organised.
I also drive a 1970 Land Rover and a number of us are struggling with the numbers on factory replacement engines and gearboxes. The answer seems to be that while there was a procedure, whether the part ever actually got stamped seems pot luck. Also updating of the log book would most likely be by hand on the Green Log Book and witnessed at a local DVLA office. I suspect many owners did not bother.
Land Rover supplies replacement chassis and I gather in the 60s it was not uncommon to get them blank with a note to "stamp the numbers on", I suspect BSA were no different.
I would be wary if the font is wrong. It was quite common to re stamp engines or remove numbers. Before opening a can of worms examine the case very carefully. I recently bought an A10 n e-bay only to be told it the frame had no numbers and the reg was missing. I canceled the deal as they had obviously been filed off.
Of course this is BSA so its quite possible the workshop had a different set of stamps to the production line
