Recruitment of candidates at entry level is based on assessment of potential to thrive as a practising lawyer doing the type of work that the firm or set of chambers usually does. Potential is demonstrated by academic achievements to date, by any relevant work experience, and by performance in selection exercises and interviews.
Many firms and sets of chambers now recruit university-blind. It may once have been the case that that recruitment decisions were made at least partly on the basis of the name of the university at which a candidate studied, but that practice has died out at the higher end of the market.
Recruitment of established practitioners is based mainly on relevant experience, and, at senior level, mainly on the ability to bring or to generate new business. Academic achievements are of relatively little importance when recruiting lateral hires who have a track record in practice. Professional life is to some extent a series of gateways. In broad terms, good A levels get you into a good university. A good degree gets you into a good job. The good first job gets you the good second job, and so on.
All organisations have a tendency to replicate themselves in their own image, so there is the controversial and sometimes unspoken element of "fit". Recruiters have to be careful here not to engage in stereotyping, and the most careful ones try not to recruit an endless series of clones who follow typical patterns.
Organisations deploy more or less sophisticated methods to aim for neutrality on sex, race, religion, disability, and so on. There is a current debate in the world of work about the extent to which DEI practices in fact operate as "stay in your lane" barriers to inclusion, and are more about virtue-signalling and the favouring of some privileged groups over others rather than real diversity.
Regardless of that debate, note the persistent impact of socio-economic inequality on professional opportunities. When I started at the Bar in the 1980s, the profession was mainly white, male, and posh. It became less white, much less male, and a fair bit less posh, but, along with all other graduate professions, it is becoming a bit posher again because of the cost of training and the Bank of Mum and Dad factor. Things are even worse in journalism, acting, and other professions where it's hard to get in unless you have money behind you. At least the law offers some scholarships.
I saw somewhere, but cannot now find, an article that suggested that recruiting university-blind might result in reinforcement of the traditional recruiting patterns, because candidates from the most competitive universities tend to out-perform candidates from less competitive universities, and the recruiters can't then say "Well, X did very well considering that she/he went to that crappy school and that dodgy university, whereas Y has cruised through Winchester and New College without breaking a sweat", because the recruiters don't have the info to do so.
I don't know what the answer is here. I think that, on balance, university-blind is best. A not so very secret secret of UK life is that the experience of Oxbridge, if enjoyed, is so intense and enjoyable that Oxbridge graduates have a tendency to flock together because they have that shared experience, and perhaps to some extent a similarity of approaching tasks, having been schooled in organisations which have for centuries been training people to be members of the learned professions, the senior civil service, the top end of the media, and so on. Other hot-house institutions such as UCL, LSE, TCD, Insead, and so on can produce similar effects. I think that most people are honest and fair minded, but humans are humans, and we can all be subject to biases of which we may not be fully aware.