Ive used EP90, or EP 80- 90 gear oil, for 30 years with zero problems. it takes 7/8ths of an imperial Pint.
This is what BSA ask for in the Manual. Do not use engine oil in the gearbox, it wont do any harm, but it will be better with real gear oil which is formulated with cats piss so that it doesnt squeeze out under the tooth point loads. The smell also helps pin point leaks.

Morris GF20 -50 is good oil for the motor.

Automatic transmission fluid, ATF, keeps the clutch happy and greatly reduces cold start stickyness.