@Arewedone
Using LSE as a fall back can be risky. A lot depends on subject. Law, Accountancy, economics, Finance, International Relations, PPE etc will be very oversubscribed. Only about a quarter of LSE students are from the UK, and in many places the LSE is seen as just as prestigious, if not more. Even in the UK, LSE may be preferred by Londoners, ethnic minorities or others wondering whether they would "fit" at Oxbridge, and those seeking degrees with particular slants: urban geography, international law, mathematical economics etc.
DS read economics, which is very competitive. The good advice he was given was to apply for the four "top" courses (Cambridge, LSE, UCL, Warwick) and accept any he was offered, and if he was was not offered any, to take a gap year and reapply. He was rejected by UCL, Warwick and Cambridge but got an offer from LSE at the end of March. He knows someone who received four rejections first time round, but a Cambridge offer, and LSE rejection, on reapplication. DS was a strong applicant (five A levels including FM and a humanity with a 4xA* prediction, plus a language at the equivalent of AS and a genuine interest in his subject) but there are plenty of other strong applicants out there, including from overseas.
LSE also recruit in a different way. They don't interview, so the PS becomes very important. They will be fishing from the same pool of well qualified students, but inevitably will select different applicants. Our observation is that those successful at Cambridge tended to be more confident, so presumably better at interview.
Which university provides the better education will be partly down to fit. DS is absolutely absorbed in his subject (he has just finished the first year of a PhD at a good US University) and found plenty of soul mates, both on his course but also amongst other UGs and PGs. The student population is scattered which ironically means that a lot of the social life is centred on the campus and he belonged to several societies and played in a casual 7 a side league. He does not drink, and clubbing did not seem to dominate student social life. I don't think he would have attended May Balls or have stepped foot inside a punt, so there was not a sense he was missing anything.
His education was fantastic, and if starting again I think he would consider LSE as his top choice. There was lots of scope to flex his course to match his interests, lots of supra-curricular stuff available, good academic pastoral care, including academics who went out of their way to support keen UGs, write references, even picking up the phone to support PhD applications. He was offered a place at Oxford for his Masters but saw no reason to move. Over half of his Masters coursemates seem to have gone on to PhDs and his LSE alumni network seems to now stretch across most major economics departments in the US.
That said, DS knew several who would have been more comfortable on a campus like Warwick. LSE is urban and you need to be a self starter, but for the right student it can be a fantastic learning experience.