I heard this Times Uni ranking being discussed on LBC radio and the woman who compiled it was saying that Oxford and Cambridge are still obviously more of an elite brand in the U.K. and internationally, but the reason Durham, LSE and St Andrews are above them in this table is that they scored better for student satisfaction.
I was amazed because some years ago, LSE was notorious for poor student satisfaction at undergrad level. It's a mainly postgrad institution anyway, with something like 80% overseas students, living all over the place. Students felt like staff were unsupportive and there was not much of a social hub at the main uni site. It's the reason my DS and his friends didn't go there!
Maybe LSE has worked really hard to combat their poor student satisfaction reputation? Sounds like it's working?
In the Complete Uni Guide, Cambridge is no. 1, and Oxford No. 2.
In the QS Global Uni Rankings, it's only ever Cambridge, Oxford, Imperial and UCL that feature in the top 10, alongside Harvard, Stanford, MIT and ETH Zurich.
In these international rankings, LSE comes quite low (about 50th), Durham lower than that and St Andrews is about 100. Edinburgh and Manchester do better than LSE in the QS Global Uni Rankings.
The other thing is that St Andrews is tiny compared to UCL or Oxbridge and LSE is also small at undergrad level, relatively speaking. So in rare cases of students getting in to Oxford and not St Andrews, it's more likely to be a matter if quotas and limited places rather than anything else. It's the same with Edinburgh, which has quotas for Scottish students, internationals, etc.