How big is the gap between Oxbridge and LSE, UCL etc for law?

I'm looking at studying law hopefully at a top uni inlcuding applying to Cambridge. Obviously I know Oxbridge tends to be the best but I'm looking to gain perpective on to what extent a law degree from Oxbridge sets you apart from the next top unis (like UCL, LSE, Durham). In particular at the moment I'd love to become a barrister at a high end chambers. Is Oxbridge only a slight advantage or would that be highly difficult to achieve without an Oxbridge degree even at a place like UCL or LSE. Thanks