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I wouldn't entirely say "not at all". I think university does play a role still, though obviously it wouldn't in the USA where nobody knows anything about Australia

However, I don't have much evidence for this - I don't really know any lawyers. I only say this because I've been to a few group shortlist interviews for legal volunteering work, and every time, out of on average 15 shortlisted people, at most one would be from Deakin - the rest were from UoM or Monash, and there was never anybody from RMIT or VU.
That said, university is mostly about your first legal job - that's when recruiters would care most, because they don't really have much else to go by. But once you get some experience, ANY experience, doesn't necessarily have to be at a top or mid tier firm, then your university (and even marks) become pretty much irrelevant.
I would also disagree with the subjects comment. I think you just need to be excellent at English. My top 6 was English, methods, chemistry, German, music and business management and I'm not finding law exceptionally difficult.