Rankings do not matter. I doubt that any law firm or barristers' chambers pays much attention to them. My chambers pays the rankings no heed. The rankings are compiled using various crude measurements, some of which are just number crunches, and some of which are subjective. This forum tends to obsess about rankings, but they really aren't scientific or particularly meaningful.
In very broad terms, UK universities might perhaps be seen as falling into four or maybe five tiers, such as -
2.
the big London universities (the largest constituent universities and colleges of the Federal University of London, plus Imperial);
3.
the rest of the Russell Group;
4.
non Russell Group universities established as universities;
5.
new universities that started as polytechnics.
But even that is a simplification. For example Oxford Brookes, a new university, is well thought of in a number of fields, and some would rank, say, Durham above, say, UCL and LSE for law, while others wouldn't. Tiers 2 and 3 above could perhaps be merged into one tier.
There is a self-perpetuating process by which some universities attract the most academically successful and promising students and get to pick and choose from those students. Those universities may also have the resources to attract distinguished academic staff, although good academics are found in many universities.
All this results in graduates of those universities being competitive when applying for jobs that require high academic standards, such as jobs in commercial law. Those graduates are hired not because they went to university X or Y but because of their individual qualities. They tend to be people who have done well in all or most levels of the UK education system.
Some law firms and barristers' chambers assess candidates university-blind. They don't do a sift on the basis of "let's reject everyone who did not go to a university ranked in the top ten of some arbitrary list".
As between Reading and Surrey, choose the one you like best and do not worry about perceived prestige. Good luck!