| Yep....You don't need a spare for the MOT, but if you do have one it has to be legal.
As to the tyre on the car there are rules (that I don't know the full details of) that state that the tyre has to have a minimum tread depth over a certain percentage of it's width, I can only assume that the tyre on your wifes car is legal even if it's bald on the inside edge.
It's probably a borderline pass though, and I would change it ASAP (and get the tracking looked at). | 
Jools
Threaded Mode