First off thanks for the help and kind words.
With direction and encouragement from Paul Jameson I've managed to track down the original registration number (in Derby)!
It's been a fun journey so far, and it involved going to Derby Council archives at Matlock, which was a nice ride on a lovely day.
Chatting with Paul, I realised the info I'd gathered DID include the reg number but I just couldn't see it. Thanks Paul!
And here's where the bike was sold, waaaay back in 1939, Palins Motors, Osmaston Road in Derby.
This pic circa 1930 btw.
NON SUFFICIT ORBIS