If you've just changed the pads and discs (ie, not fitted JCW
brakes) then you won't have changed the size. So you need whatever size wheels your car needs - use the wheel FAQ
to work out what your model Mini needs. Typically if you have a 2nd Gen Cooper S you need at least 16" wheels, whereas most other Minis can use 15".
Having a second set of wheels means you can swap the wheels yourself at home, whereas if you just only winter tyres, you have to pay for them to be swapped twice a year on and off your current wheels. So a complete set of wheels with winter tyres is best, if you can manage it, plus then you can choose to change them whenever you want, rather than when you can get to the tyre place.
The other thing to decide is whether you want the winter tyres to be only for emergency use when it has snowed or is going to - in that case you can use smaller, cheaper winter tyres and cheap wheels. But that will be throwing away lots of the benefit of winter tyres, like their better wet weather braking and grip on ice.
The other plan is to put the winter tyres on for the whole winter (I reckon, 1 Dec to 1 Apr) in which case you might want bigger winter tyres to give you better grip when there isn't snow and ice around, plus you may then want wheels you are happy to see on the car for several months.
Plan A (emergency use only) if you don't have a 2nd Gen Cooper S would be some 15" steel wheels or secondhand 15" alloys (similar price) with 175/65-15 winter tyres. All up that should cost £400-ish, or about £600-ish if you bought them new from a Mini dealer. Or there's a guy on totalMini
selling a complete set like this for £200 - but it's snowed today, so they're probably sold!
Plan B (all winter use), which would also be the choice for a 2nd Gen Cooper S would be some second-hand 16" or 17" alloys with 195/55-16 of 205/45-17 winter tyres. This would 'only' cost £500-600 including the wheels so it's not much more than Plan A. I spent £250 for some very nice Pace spokes and £550 for some superb Conti 17" winters, so that is the top end of the price range for the absolute dog's whatsits.
It seems expensive when you first pay out but remember the winter tyres mean you won't wear out your summer tyres as fast, so you get some of the money back, in a sort-of way.
And then you get to drive in comfort - today I was that lunatic (according to all the other drivers) who could sit in the slushy lane 2 comfortably doing 70mph on my winters while everyone else was queuing in lane 1 behind the people who think 40mph is the maximum safe speed if you can just see
snow - there wasn't actually any snow lying in lane 1, but that's not relevant to the super-cautious.