If you mean by tracking the front toe-in adjustment, you can actually do it yourself
with a tape measure with reasonable accuracy, as can any alignment shop (although
they use slightly more sophisticated equipment).
Mine was off by 1/4" in the front when the car was new, and after I adjusted it,
it has stayed dead-on ever since. The rear toe in is a bit harder to adjust in the MINI,
and the camber is not adjustable at all in stock form for the front nor the rear.
Changing wheels and tires should not affect alignment unless they dropped the car off
the lift or ran it into a curb before or after the process. Lowering the car will necessitate
getting adjustable rear control arms to reduce the excessive negative rear camber that
goes along with lowering it, and then re-setting the rear toe-in. If the front camber if off, something is bent, and suspension parts need to be replaced.
The fancy equipment BMW/MINI uses for checking alignment is state-of-the art, and
very accurate (as well as expensive and time consuming), but it's a bit pointless
with a car that has very limited alignment adjustment options.