C4556070 was the British census number for WM20.46561. This was a contract C9310 bike (9.000 motorcycles, census numbers C4553722 - 4562721).
BSA WM20.46561 was despatched from the factory on 07/10/1941, destination "Ministry of Supply, London SW1". Quite a few C9310 bikes were redirected to Australia, where they received a new (Australian) census number. Here's another example from the same contract: British census number C4558184 - frame number WM20.48675 - Australian census number 80034. You would have to go through the AWM ledgers to find your bike Barry...