I get my engine oil for all my cars from Euro car parts, their always having deals, buy one get one free, 50% off, a free oil filter, I buy it when it's on offer and keep it in stock. I use their QX brand 5w 40 fully synthetic with a Mann filter which comes with a sump plug washer and housing seal. If you haven't done them before it's wise to replace the two small rubber seals at the base of the filter housing (only available from BMW p/n B11.42.7.549.573 the last time I bought them £1.59 each)
Gearbox I've put Amsoil fully synthetic MTF (not sure if it's still available?) that was recommended by the US contingent back in 2012, but I did replace the gearbox oil on a friends 2.2 to cure poor gear change and Opie recommended Fuch's Titan sintofluid 75w-80 and it made a great difference.
Diff (non slip) oil Castrol Syntrax universal, bewere it stinks when it comes out!!