Prior to aquisition by Raleigh (1952 or so), BSA made everything. Even the smallest internal parts on a hub geat are stamped with the 3 rifles. Simple nuts and bolts have the mark. A BSA roadster bike might have the rifles in 200 places.
BSA was a very premium brand at a premium price. Check the catalogs. The old track guys stripped the Paramount parts off the Schwinn Paramounts for BSA and always preceded BSA with "genuine". It was as revered as Campagnolo.
The real hub gears are totally different from a Sturmey. The trigger operates "backwards" so a slack cable is first gear. When I overhauled one it seemed even better made than a concurrent Sturmey.
Except for pedals, all the BSA stuff is so durable, it is not too hard to find.
Joe B-Z Stuck in hotel room in Las Vegas for 2 months