I was initially worried about moving from Oxford to UCL for the course that I wanted to do (MSc Climate Change). I was concerned that the move would be perceived as me either not being able to, or not wanting to hack it at Oxbridge.
But when you move to a PhD, you totally change setting from educational to research. If you were asking where to go for your undergrad education I would say Oxbridge hands-down; even if the formal education isn't any better (and I don't believe it is any better), the academic standard of the accepted applicants is higher so you develop more academically. You probably know this.
But in the research setting, Oxbridge suddently has this *huge* influx of people who didn't get in the first time around and are seeking prestige, and the standard of accepted applicants drops. That's one thing. The second thing is that UCL is incredibly attractive to the worlds top researchers in a way that it isn't attractive to the world's top prospective students. Firstly it's near several other universities like Imperial, it's part of the University of London, it's down the road from the Westminster Gov. Secondly, it's just massive itself which opens up much more collaborations and possibilities for lab, compute and equipment access (in part also through economies of scale). So the research environment at UCL beats Oxbridge in a lot of rankings (which I appreciate have their problems - separate discussion): it beats Oxford here and beats Cambridge here. Interestingly in that second one DeepMind wins...
So I'm not saying you shouldn't care about prestige, but I'm saying that "Oxbridge" prestige doesn't actually hold against UCL in the world of research, which is what you'll be doing during your phd. Make sure you're not viewing things as if you'll come out after four years and just have on "PhD Cambridge / UCL" on your CV: the "prestige" of these places isn't from the institutions themselves, but from the achievements and collaborations you make while you're there that are unlocked by the institutions. And UCL is easily as competitive as Oxbridge in that regard. I think that's what sells on the CV.