As long as you attend an RG uni then you stand as good of a chance as everyone else. City firms recruit from a wide variety of universities. Institution is becoming less important. It's no longer the case that you have to be from Oxbridge to make it to the magic circle. If you are thinking about the bar, then I would suggest Oxbridge. Other than this, academics are only 50% of the picture.
Law firms will reject academically gifted candidates if they don't have evidence of extra-curriculars and people skills. So make sure to do lots of things outside of your studies when at uni (committee positions at societies, volunteering, mooting etc). I have friends at Oxbridge who haven't manage to obtain training contracts because they lack experience outside of their academics.
Anyway, if "prestige" matters to you then the obvious choices are Oxford, Cambs, LSE, UCL, Durham etc. As long as you attend a RG or top 20 uni and obtain a 2:1 minimum, then academically you should be fine.