The answer to your question depends on what type of job you seek. If you wish to become an academic, then the university you studied at may impact your career options. If you wish to practise at the top end of the Bar or in the biggest law firms, your university may be relevant to your prospects. Even when chambers and law firms recruit university-blind, they still quite often select people who have studied at well known universities. This may be because those universities equip people to be competitive in the selection processes, and in the work done at such chambers and firms. The teaching methods at Oxbridge, for example, tend to assist people who wish to become barristers in the top sets of chambers. This is not to say that you can't get into such chambers unless you went to one of the better known universities, but you will be competing for pupillage against people who have studied at those places.
Chambers and law firms do recognise that many factors can affect which university a person goes to. Socio-economic disadvantage and specific personal circumstances during the school years can narrow the choice of universities accessible to some students. Very talented candidates who have not been to well known universities can still do well, but, realistically, they face a harder road than those who have studied at more highly rated establishments.
In a great many first jobs, your degree results will be what matters, and not the name of your university, and, as you move into your second and third jobs and beyond, your university career will become of ever diminishing relevance.