Lincoln is a very nice and safe place to live but it is very small. I'm currently applying to uni and I have applied to both. Both universities are nice but I'd advise that you go and look around both of them. As for a car at uni, if you go to either of these, you wont need a car as public transport is good.
The accomodation at Nottingham is very nice. As for Lincoln I'm not sure as if I go there, I am planning to love at home. Nottingham is the more diverse city and according to a friend who studies there, the nightlife is better.