The MINI factory wheel locks is supplied with a key code card which should allow the MINI dealer to find a replacement key. Without this key code, you will need to find alternative methods of removing the lock.

You might try your local body shop - they have ways to remove locking lug nuts without the keys - something they should be used to dealing with. The lock will be destroyed, but at least you will be able to remove the wheel.

I bought a set of Craftsman ones years ago. 
They fit over the original locking nut and bite into the lug and pretty much ruin it but at least they are removed.
On weird sized ones I have hammered an old used socket on the lug and removed it, always with a bar or ratchet not an air gun slow and steady is the best method.
Try the service department at the dealership I am sure they run across this problem all the time and maybe have the correct one to use.

Working on clutch replacement 2015 Mini (wife car)run at next problem,no way to find safety socket lug nut key ,it is clover shape .Any idea how to find right key socket for lug nut on 2015. Have try advance auto and O`railly universal set up did not work on mini even master key what is for sale in harbor fright did not fit. Thankful like always for all advice's. Pred