It depends on the type of shafts.
If you have the Cooper S or early Automatic type, which are a small universal joint, with four nuts, then you can wriggle them back enough to get the engine and box out.
If you have the rubber cross and u-bolt type (most Minis to about 1974) or the usual Pot Joint type (All Minis from 1974ish) you must undo the top ball joint on one hub and support the hub. I undo the radiator side top ball joint and slide that one clear, then tilt the radiator end of the engine and trans as I lift it and pull the clutch side one out.
Withe the pot joint type you either have to drain the oil and remove the whole Pot by popping it free from the transmission, or you can remove the inner boot and pull the inner part of the Pot out. Then you will need to chase all the little greasy balls that fell on the floor and collect them up.
Whne putting the greasy balls back in, clean them up, click them in place, regrease and then wrap them in a couple of layers of cling-wrap. This will hold them in place while you reseat the inner part into the outer. You can leave the cling-wrap in there, it won't do any harm.
Cheers,