Rear discs are a pretty rare thing... A bit pointless, really, given so little braking work is done by the rears. KAD has done a conversion, but the point of it is weight saving, not increased performance.
MiniSport (in Adelaide) are KAD agents.
So, if you're after increased braking performance as the motivation for wanting rear discs, I'm sorry to inform you there's no benefit.

I understand if your idea is to reduce maintenance (I'd do it for that reason!), but it's a fair amount of cash to splash if you're just being lazy.

Only reason for it is reduced unsprung mass, which is only relevant if you've got extremely light wheels and are competing in Motorsport.
For the fronts, easiest way to to it is to pay a specialist to fit them, otherwise, you'll need to buy a full pack. It will include Disc Brake CV's (which are bigger than drum brake CV's), Disc Brake Hubs, bearings, rotors, calipers, etc.
The kit you buy will be determined by two things - wheels and budget. What size wheels do you have? If you've got 10" wheels, you can only fit Cooper S discs under them. These are 7.5" diameter - they're a very good option, well used for 45 years now. If you've got 12" or bigger wheels, you can also fit 8.4" discs under them. These discs are cheaper to buy/maintain (cheaper rotors), and were fitted to 1275LS (Clubman), Moke, and all UK Minis after 1984 (hence higher numbers produced).
Cost? You're looking at $1500+ (up to $2k) just for the S Discs, maybe as low as $1200 for 8.4" discs. These are new prices. You many find some second hand, but by the time you refurbish them, it'll end up at similar money.
There are other options, but they involve much machining, similar or higher costs, and often requiring machining rotors (so aren't as simple to maintain - buy replacement parts). A search on here will give you more info.
I'd recommend giving MiniSport a call.
