You would do well to look at the course content for each and compare them. Sometimes they can be very similar. At my university, Coventry, I'm pretty sure only a couple of modules are different between mech and motorsport, and it's basically the "motorsport version" of those modules. For instance, "instrumentation and control" for mechs and "vehicle instrumentation and control" for the motos.
You could feasibly work in any mechanically based industry with either skill. Mechanical may be broader, but you won't be forced to work in motorsport with a motorsport degree and similarly you shouldn't not be able to work in motorsport with a degree in mech. My friend did motorsport and is now a structural engineer with JCB, so there's certainly flexibility.
With regards to which university, well, whichever takes your fancy. Coventry's engineering department is fairly nice, but you need to check them out and see which you like the most. I wouldn't focus purely on which university is "higher" than others. Certain universities have better links with industry than others and employers mostly care about the degree classification you get rather than where. At Coventry we get great connections with industry and there have been McLarens parked outside more than a few times as they come to our uni to provide talks.