The speedo and oil pressure gauge are mechanical and have nothing to do with the electrics.
It's probably a dodgy fuse or poor contacts at the fuse block on the A3/A4 fuse. As well as the indicators, temp and fuel gauge not working the wipers and most importantly your stop lights won't be working either as they are all supplied via the same fuse. This only has supply when the ignition is on.
The other fuse A1/A2 supplies the horn and interior light. This has supply all the time.
Despite what others say the lights are not fused.
The ignition coil and fuel pump are supplied via the ign sw but are not fused either.
Mini electrics are pretty basic and not everything is fused like more modern cars.
If the generator is charging OK there is nothing wrong with the voltage regulator. This won't make your indicators work if you replace it. New ones are around the $100 mark so there's no point replacing it. If it aint broke don't fix it!
Hope this helps
RonR