If your nan gets the high rate mobility part of DLA, there is a chance that your nan could get a Motability car but would have to give up most or all of that part of her DLA to get it... it isn't something extra on top unfortunately, so it works out about £160 ish a month that she will lose. You used to be able to apply if you got middle rate mobility, (I'm not sure if you still can) but most cars cost more than that per month.
It would technically be your nan's car, but she could have your mum as a named driver so she is covered for insurance etc. It might be worth having a look at the Motability website for more information. Here