Hi Richard,
Strange one here, I have never ever come across any reference to the British Forces using BSA Bantams, and being a one time Bantam owner and other BSA's in the past, I am sure I would have remembered. If you are talking about the late Forties / early Fifties period, there was no MoD then, vehicles were procured by the Ministry of Supply. If MoS bought them, they might have been used in some other government department, perhaps these are the GPO telegram bikes (?), I don't know if GPO vehicles came via MoS at that time. That is quite a large number, 6,500, so if the army or navy did have any I am sure we would have seen surplus ones come to light before. About that time there were postwar model James lightweight m/cycles used in the Forces.
Will keep on looking and see if any reference turns up.
regards, Richard