MINI COOPER S 2010 is churning out blue smoke...HELP

You're burning oil. There's a few things that can cause this. Piston rings could be worn, sometimes a head gasket depending on where the oil galleries are and valve stem seals. Get the engine compression tested to make sure its sealing properly. If compression is good its likely not rings or a head gasket so it must be valve stem seals or making it's way into the intake manifold. Get the PCV valve checked if you have excessive oil in the intake. Beyond that, stem seals which means the engine has to come apart. Unfortunately once your car starts shooting the blues there isn't usually a cheap fix. If you want to minimize the burning you can use a thicker oil (if the car takes 5w20 try 5w30 for example) in conjunction with some sort of oil treatment that helps keep the smoke down. If you try that and it keeps the smoke down or ideally stops it the next time you head for the MOT get the car nice and hot beforehand if possible. The catalytic converter on the exhaust needs to be hot to clean your exhaust properly and pass emissions. If you're lucky you'll squeak by without breaking the bank. So first get a compression test done and the PCV valve looked at. From there you either go with likely a very costly repair or run thicker oil with a treatment in it, get the car hot and cross your fingers. I hope it ends up being a minor issue. Good luck!