BEng Mechanical Engineering -> MSc Aerospace Engineering is the route I'm actually going on (I'm in the first year, but hey, it's a plan, right?)
You'll have to check entry requirements to be entirely sure, but every university offering MSc Aerospace seems to be willing to accept BEng/BSc Mech Eng. It's basically the same sort of engineering, so it will qualify you to study that Master's degree.
Mech Eng is a broader, but not too broad, sort of engineering. The first 2 years of Mech and Aero would probably be the same, and there would only be minor differences in the third year. Mech Eng will probably be the best route as you will gain the most experience in a variety of fields, then you can specialise in Aerospace afterwards. The courses are otherwise pretty much identical with most universities. You might get a few more aerospace tailored projects if you did find BEng Aero, but I've done an aerospace report already with Mech.
One thing to be aware of, though: Not all the MScs are accredited, and this affects more than just BEng/MSc, but just check that the university is properly accredited. Someone like "IMechE" will be perfect. That way, you'll be able to become a chartered engineer in the future with the least hassle. It's possible to become chartered in other ways, but it might mean more work from you. You'll probably also be accredited by "RAeS" and "IMechE" for the MSc. I spoke to IMechE and they were happy to accredit me and said I would be able to become chartered with BEng Mech+MSc Aero