In all the years of mini ownership, and I used a mini for a daily for about 12 years, I only ever 'broke down' twice. Once was actually my own fault, as I had gone out to the work car park to check a brake pipe I had replaced, found a small leak and then stripped it in trying to tighten it up (I had cross threaded the pipe). The other time was caused by a wire chafing through on the clutch master cylinder body and shorting out.
I would say that in theory a mini should be more 'reliable' than alot of modern cars, because they can still soldier on with things that are not quite right, usually giving you warning that you need to sort it asap! This would be especially true on a carb mini, which has no sensors or engine management.
Slight exception is the injection cars, I wouldn't say that they are particularly more likely to let you down, just that in theory there is more chance that something *could* bring you to a stop - basically because they have a few sensors and an ECU - the Rover ECU in the mini is very reliable though. As an example, the crankshaft sensor failing will stop the car dead. Having said that I have never encountered that problem.
On a carb mini, the only thing that would likely catch you out would be an ignition fault - eg failed distributor points, or wet leads (a bit of wd40 will sort that) However this should be pretty unlikely on a well maintained carb mini.
So in summary, I say minis are very reliable. The mechanicals in particular are extremely reliable, only potentially let down by the chance of an ancillary part failing, and even then it's not common and not always going to bring to you a halt. A few spares, a basic tool kit and a little knowledge and there's very little that will leave you stranded.
I'm now laughing because I am just recalling a 1986 mayfair (so a carb car) I had for a few years. For some reason it would not start on the key, but it would bump start no problem. So I would push it then jump in and bump start it, every morning. Then, half way to work I would pull over, and whilst leaning round the car, operate the clutch pedal and use a spanner to bleed the clutch slave cylinder - there was a leak so it would fill with air very quickly, until the clutch wouldn't disengage! Can't imagine being able to limp a modern car along like that !
I should point out this was a very rough car, which I wasn't really looking after very well 
Incidentally - when you compare Rover with Austin, bear in mind that the real comparison is injection / carb - Rover made carb minis up until the SPi in 1992.
The real question is does the thought of getting your sleeves up and having a more involved role with the maintenance of your car worry you or do you look forward to that opportunity? If you are prepared to get involved with the car, I think you would find it rewarding and lots of fun!
Edit - my personal view is that the SPi minis made from around 92-96 are one of the best. The SPi system works very well and is pretty bulletproof if looked after and armed with a bit of knowledge.
Edited by carlukemini, 11 January 2015 - 10:59 PM.