Regarding the prestige of the program, I would add that it also really depends where you want to work. I am originally from a middle eastern country and I don’t think imperial or LSE will be that known in comparison to Oxford or Cambridge. However, this isn’t true if you were to work in the US- LSE and Imperial are both pretty known since ranking is much more prominent.
What I said above is probably true for career in industry and not research.
What I said above is probably true for career in industry and not research.