would sort of rule out 02 sensor as when the car is cold the 02 sensors are switched off and have no effect on the running of the car, the only way you stand a chance is by having a live data scanner that allows you to view each cylinders smooth running and misfire count also it give mg's of fuel demand on each injector with this car tel what cylinder for a start and whether is a fuelling issue or a electronics issue,, autocom cdp plus off ebay is by far cheapest and best for money, bmw mini software insta d bt a lot harder to run and work with autocom cdp is by far easier and covers all cars on the road as well does everything to be honest,
so once you can read what cylinder is misfiring then you can start moving plugs and coils and injectors around to see if it moves to another cylinder if does then thats the issue if dont then need compression test a must do job,
most common issues are timing chain stretch after 60.000 miles they all need one, also on cooper s turbo engine the high pressure fuel pump can fail once again need diagnostics to read it, also before replacing this pump its worth replacing the 12v in tank pump as common to fail after 70k and will cause same sort of issues ie when engine demands more fuel to rev the pump is starved,
