According to my Panther reference material:
P & M was Burmans last gearbox customer. Even though they were long declared obsolete, and no longer used by other marques, Burman continued to supply BAP boxes to P & M through to approximately 1962 (possibly from running down their remaining stock towards the end). Burman finally scrapped their gearbox making facilities around 1962-3 I think.
Between 1963 - 5, while under receivership, P & M continued to assemble M120s, but with AMC boxes (there is some suggestion that they may have also fitted reconditioned BAPs). The AMC boxes ultimately proved too expensive - 4 times more than the old Burman box as you say. P & M motorcycle production officially ended in 1966, although a few further bikes were assembled from spares.
Regarding my 1960 M120 - the POC machine dating registrar has confirmed that it is highly likely the frame and engine left the factory together (numbers never matched, but mine are very close, thus likely to be "original", not that I care). My magdyno is the correct model but from 1954. Similarly, my gearbox is the correct model, but is from 1956. If I recall correctly from 1958 or 9 onwards, the BAPs were oil filled, thus had an oil seal and drain plug. My BAP is the earlier box filled with semi-fluid grease.
The BA box on my 1950 Ariel is internally very similar to the BAP I think, but has the external clutch lifter arm. I think the mainshaft is a different length also.