Curiously I have just renovated a 1947 vintage hay mower which has a Villiers 2 stroke petrol engine driving a steel worm and brass wheel transmission.
The manual states to use SAE 30 oil as the 2 stroke mixer, and crucially says to use the same grade of oil in the gear box.
When I stripped it the gear box definitely had thick gear oil in it. Something the manufacturer warned against. Mainly because the internals are splash lubricated.
I would have put money on using a specific gear oil but Pennine lubricants echoed exactly what the engine manufacturer said in 1947.
Bough a 5 litre can of classic car SAE 30 for 24 quid, job done.
Mick