Because you asked, I run Classic Car Motor Oil in four of my MGs and Mobil 1 in another. I highly recommend against changing from dino to synthetic in a higher mileage engine. You risk serious issues if the synthetic knocks old chunks of crud loose within the oil passageways.
To address a couple other comments in the thread, Castrol GTX has reformulated several times over the years. It no longer has the level of zinc that is considered necessary for our old engines. Running it won't create problems overnight, but in theory will accelerate wear in the valve train. I suspect Paul doesn't actually run racing oil in his engine, as that would be totally inappropriate for street use, but more likely something like VR1 which says "racing oil" on the bottle. These type of oils aren't really racing oils, they just use the term to sell oil to sports car owners. VR1 does have a decent level of protection however and it isn't a synthetic.
For other fluids, I run Royal Purple in all my M.G. rear ends. The MGA, MGBGT and MGC gearboxes get Castrol 20W-50 while the MGB gets SAE30-ND. Brand isn't really critical here, in my opinion.
You can run synthetic in a gearbox, many use Redline. However do NOT use it in a Laycock overdrive because the friction modifiers will be too slippery for the clutch cone.
You should really buy a repair manual, as it lists all the fluid types and quantities necessary. Some of the fluids will be outdated of course, but the general type and how much to use will be covered. It also shows procedures, torque specs, etc which are all things you should have on hand while working on your car.
Paul, as a side note, your TF should use MT-90 gear oil or equivalent in the gearbox, not engine oil.