'FAU' with a 'C' suffix is a 1965 Nottinghamshire series so the bike is likely to have had a post-war registration...It will be a job to find it in the 'Key Cards' though unless they used the engine number.
These 31/BW/.. numbers do seem to be a rebuild number presumably stamped on a new replacement frame or a frame with a previous number filled off, is there a flattening or any file marks under the number?
31/BW/24 is listed as a frame number in the post war key cards.