This is a daft question. Firstly, how on Earth could projects or supervisors be "equal"? Human beings are not quarks, they are all very different. Secondly, it really doesn't matter at all which research institution you go to, or what research you do, as long as the research you get involved with is recognised as first class in every respect. The most successful UK Astronomer I know got his PhD from Leicester. He was lucky, or wise enough, to find a "turned on" supervisor and a great project. So talk to the undergraduate lecturers you respect, from long years of involvement with them. See if they can offer you a good research position, or "know someone".