If you're thinking of buying a new one you could use one of the combo British Standard/Whitworth tool kits, or just a combo of SAE/Metrics. An old BSA only uses about 5 different sizes on the whole bike.
I have a set of BA/Whitworth spanners somewhere.
So to answer your question, they are neither metric nor SAE. They are British Standard or Whitworth, depending on the age of the machine.
before I hurt my knee I was helping to tear down for overhaul a 1972(?) 650 engine from an old chopper that scrdvr bought( all there). he has the tools and special tools. Tho one gear puller is a PITA to use.