I have a set of new snow tires mounted on my 06/16" rims (stock Mini rims)and I am being told they will not work on my 2012. TPMS is different and the wheels won't fit. I sold the 06 and bought the new 2012 Cooper S a month ago...Help...

 

As long as you check your tire preasure manually...just live with the tire preasure light...a piece of electrical tape to cover it...
if the shop will not/cannot put the rims on your car due to tppms not being present, just do it at home...there is a bit of a liability/legal issue with tppms sensors, but the rim must ne compatible...the 2006 rims are not to my knowledge compatible ....your old car montoring wheel rpms.....the new system uses actual psi in the tire....more accurate, but more $$ and more complexity.

 

The above will only work if they physically clear the brake rotors/calipers....
If they don't...
You have rims/tires for sale...i do not think spacers are really an option.
Good luck!!
PS to my knowledge there was only one 16' gen1 rim that would clear the gen1 JCW brakes....which are the same size as your gen2 brakes....