Grace's Guide is the place to go to learn about Brit companies. This is what B.S.A. Tools Limited was doing in 1939 - making small machinery for automotive/aircraft industries. I would say you have a wrench that was supplied as part of a tool kit for one of these machines.
"B.S.A. Tools Ltd., Sparkbrook, Birmingham. The specialities of this firm are single-spindle automatic screw machines in five sizes with every refinement in design for quick production.
They also offer B.S.A. Acme-Gridley multiple-spindle bar machines, single-spindle automatic chucking machines, multi-tool production lathes and the like. Centreless grinding machines and external lapping and honing machines of very fine quality are also made. The range of all tools includes milling cutters, taps and dies, twist drills, reamers, hollow mills, counterbores, etc. "
https://www.gracesguide.co.uk/BSA:_Tools