I am not a lawyer, but was running my mother's affairs for the past decade and had dealings with several departments of her local law firm, plus some litigation using a specialist London firm.
Degrees came from all sorts of places. The ones I looked up including two newish graduates: UWE and Bournemouth; slightly older graduates from: Bournemouth, Aberystwyth and Kent. And a couple at least, from an older generation, with no degrees at all.
Some were very good, some not so. Thing is that a lot of the work is relatively routine, and other skills come into play. One had picked up expertise in sheltered housing leases, and was brilliant with my mother who was, naturally, unhappy about having to leave her previous home. Another was able to give me lots of good general advice about employing a carer directly, based on long experience with the problems local small employers had faced. Its not just about what the law says, it is about finding solutions. Ditto with a tenant issue, and currently with my mother's probate. In many areas of law, especially in local practices, it is as much about communication, common sense, wisdom and experience.
Getting your foot in the door might be that much tougher if you don't have a name degree. However if someone makes a real effort to gain complementary skills via volunteering or internships, they might find themselves well up the queue when it comes to the regional firms.