I have been having this same problem with my 2007 mini cooper base model. If i have my car running but not driving for around ... 10-15 minutes ... as soon as i start driving it shoots out what looks like WHITE smoke out of my tailpipe....and its a LOT of smoke.

 

Mini Cooper s oil needs to be checked every 1000 miles unless you want an expensive timing chain bill .When the oil gets low the engine over heats .I have just paid £1120 to have mine fixed .Be warned !!