Have a recently acquired a '59 Magnette which is mechanically the same as an MGA, but built up into a luxury sedan model. Relatively few were built and exist today.
No owner's manual but I do have the shop manual. It specifies the same oil for use in the transmission as for the engine (oils separated by the dry clutch & bell housing and don't mix).
My immediate thought is that a heavier oil intended for gears would be better, but who am I to argue with the folks at Abingdon on Thames in the late 1950s. Back in 1965, after using a good grade of SAE 30 non-detergent engine oil for several years in an identical car, I experienced a transmission failure. Ball cage on the input shaft bearing had disintegrated and also found wear on the layshaft.
What are other's using, comments welcome, & thanks ....