Nottingham has the best reputation and from a lot of personal experience (I don't go to Nottingham uni though), I can tell you that it's a very nice city, with a superb bus service, a tram that was finished in 04/05, and lots to do (ice rink, decent cinemas, theatres, lots of shops, "Rock City" etc). Go out into the suburbs of the city and there are some lovely parks. Even near the centre of the city, there's Wollaton Hall, which is a really nice place to go and relax. Pretty much centre of England so you can get to most places with relative ease. Added to that, it's the uni with the Economics department that has the highest reputation out of the 3 you've mentioned. The city has a stigma of being pretty crime-ridden, but avoid the bad areas (St Ann's, Aspley etc.) and you're fine.
Lancaster is in the middle of nowhere and has the lowest rep, and I have no clue about Exeter. You should visit all of them though.