Many people have their own opinion of which oil is best for the Mini.Matic.
One thing that can be agreed is that the oil needs to be replaced regularly for the transmission to perform.
The A-Series engine requires a densor oil compared to modern vehicles. It is recommended that you buy non-synthetic oils because most synthetic oils are designed for modern engines that can cope with lower viscocities.
"Oil pressure is directly related to viscosity and it is therefore not the difference between synthetic and non-synthetic oils that causes the difference in pressure, more the fact that modern synthetic oils are of a lower viscosity than the A series engine was originally designed for.
Whilst the lower viscosity of some synthetic oils may lead to reduced oil pressure, it does not necessarily mean reduced protection. However, owners who are nervous of a reduction in pressure should consider using Castrol Formula RS 10W-60 which will give rapid circulation when cold but will retain a substantial viscosity when up to its full working temperature." - Castrol (UK) Ltd
What you do want to avoid in the Mini.Matic is any form of "friction modified" oil, because of the effect this has on the numerous wet friction elements (2 clutches, 3 bands) in the gearbox.
In fact if you want to spend money on really good oil, a thicker motorcycle oil designed for bikes which mostly have wet clutches is probably the go.
The parts in the Mini gearbox that suffer most are the large crownwheel bearings and the baulk rings in the synchro hubs.
What you want to use is:
Valvaline 20W-50 -- non-synthetic Multigrade oil only
SAE viscosity gradings SAE 10W/30 SAE 10W/40 SAE 20W/50
Further reading: Link: Motor Oil Viscosity Grades
Link: Wiki: Motor Oil
Link: Engine Oil Bible