The VMCC library holds some original BSA factory ledgers that may be of help, but quite a few 1939-40 BSA's where sold and then requisitioned by the UK government and its a matter of opinion whether these count as "Military" bikes.
The last KM frame listed in O&E appears to be KM20 5734 delivered to the war office? in Sept 1939? and I would guess this information came from the BSA ledgers.
6092 is probably not a British War Department machine but could have been supplied to another nation (Ireland and The Netherlands were large customers during 1939).
The best thing is to e-mail or phone the VMCC library. Unless you're very unlucky with a missing reference, they'll be able to tell you original dispatch date and destination (in the case of UK machines, the dealer but overseas usually just an importer).
It won't cost you anything to ask and will end the speculation.