Hi Paul, first of all the machine in the photographs you show is without doubt a BSA, In terms of BSA model years, the A10 was introduced in year 1950. The A10 used a re-designed and improved version of the A7 engine with the design work carried out by Bert Hopwood who had been enticed away from Norton in 1948. After the Second World War only 350 cc and 500 cc overhead valve models were continued. The machine is most likely an A7 and fits in with the 1940s, that said your fathers clothing looks to be a little later than the 40s riding gear people used at the time, however your father looks a cool dude and a bit of an adventurer. The BSA owners club can assist, using the registration they will be able to telly you the model, year and were the bike was despatched to, unfortunately I don't have any contacts within the BSA owners club to reach out to assist, as I mentioned please go through the owners club who I am sure will identify the bike for you.
It's an A7 or A10, of course. Everything that follows should be prefaced with "as far as I know". It's definitely not an "Ariel" front brake, it's the BSA full-width front hub that was introduced in 1958. The headlamp nacelle is the later type, also a 1958 change. Can't see if it has an alloy head, so I can't tell if it's a Shooting Star or Super Rocket or the standard iron head A7/A10 engine. However, it does have the blade type front mudguard, not the valanced one, which suggests Shooting Star or Super Rocket. I don't think there were any notable changes between 1958 and the end of production in 1962. So... it's a 1958-on BSA A7 or A10, probably a Shooting Star or a Super Rocket. Of course, it could be a later front end on an early bike.
Paul Brockwell Welcome to the family. Yes i would say A10. But did ponder on it being a Royal Enfield Minor Meteor. Great old photos thanks for sharing. Joe .