I believe that Minis normally have dual mass flywheels. The (unfortunate) difference between the dual mass and single mass flywheels is that single mass flywheels are able to be resurfaced whereas the DMFs are not. They have to be replaced. Resurfacing is cheaper than complete replacement. It is possible the "One" version of the Mini has a single mass flywheel - check with your dealership to be sure.
My clutch started to go at around 56,000km. However, I complained to the dealership about a rattling noise when the car was in neutral with the clutch engaged (pedal up) when the car had about 20,000km on it. I was constantly told this noise was normal. Once the car hit 56,000 km, the clutch and flywheel were slipping badly. Furthermore, owing to a lack of available service appointments, I had to drive the car for about 7 days with a badly slipping clutch. The end result was an excessive amount of heat damage to the flywheel.
When the clutch and DMF were replaced, the car was out of warranty. I wrote to BMW/Mini and explained that a complete clutch and DMF replacement before 100,000km was not normal. In the end, Mini agreed to cover the cost of the flywheel, but I paid for the clutch and labour.
Even with reimbursement for the flywheel, this was very expensive.