I've applied to both and got offers from both.
I don't think ranking matters for these two schools as they are too close to each other.
Oxford accepts candidates from much diverse professional background so the calibre of students be less consistent than that of Judge students. But instead, Oxford accepts c. 50% more people so in terms of absolute numbers you would probably have similar number of quality students in both schools (whichever sector you are looking at).
I chose Oxford because: 1. the faculty's strength in Finance / PE and Finance Lab (didn't want to go to LBS because I wanted to part of a wider university and live away from London for a year!); 2. larger alumni base; 3. better established extra-curricular activities (e.g. finance lab, creative destruction lab, oxford union debating club); 3. larger number and amount of scholarships offered and 4. multidisciplinary curriculum that connects different courses under themes.
I saw from a couple of forums that Said alumni are less approachable. I thought the same at the start of the process but to be fair it really depended on whom you contact. Some Cambridge alumni were really helpful and offered their time for a call but some just didn't respond. It was same with Oxford alumni - some offered their time and some didn't. My tip for reaching out to alumni would be to find someone whom you have a few things in common (e.g. residence, industry, university).
I hope this comment is helpful for those who are going through the process that I went through! (I think I read literally all Said vs Judge post available online

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