I think the cause of the biggest variations in rankings comes from the weight they give to research (strong research Universities are not always strong teaching Universities) and to student satisfaction (the Guardian is big on this, which can lead to some very strange results). The LSE in particular jumps all over the place. It notoriously gets very low student satisfaction results (DS told me that there was a badge of honour associated with the fact they often came bottom, which may skew responses to the questionnaire.) It also, obviously does not offer STEM, or do STEM research. Per subject tables are probably more useful when it comes to outliers (Dundee - computer science, Northumbria - product design, Surrey - automotive engineering, Bournemouth - costume design, LSE - social policy, Cardiff - Journalism etc) and even then is can come down to the type of course. You really don't want to read economics at LSE if you don't enjoy maths.
DS is now studying for an econ PhD is the States, and there seems to be an informal consensus on which Economics Departments are strongest. They change, as staff come and go, if the University invests in the department or perhaps if they land a stream of research money. He says that LSE is currently on the up, though did not say why, though I understand that there is a slight post-lockdown reversal of the traditional brain drain from Europe to the US. His programme (six years!) normally takes a couple from British universities each year, seemingly evenly split between LSE and Oxford. Prestigious depends on individual judgement, but our observation is that internationally LSE opens as many doors as anywhere else in the UK.
Toddlingturtle's posts are interesting. I am sure plenty of London kids want to leave London and plenty of schools will recommend it. DC were at a high achieving academic private school in Central London. Generally the focus was the course, so normally the aim was Oxbridge with London as the fallback. From the school website the breakdown for 2022 was:
Oxford (42)
Cambridge (35)
UCL (17)
Imperial (15)
LSE (11)
Other UK (56)
EU (5)
US (35)
Of the "other" a fair number will be medicine, or more specialist Universities like the Courtauld. Edinburgh was always popular, especially for those who held EU passports. (I am not sure how they handle gap years/reapplications.) There were also some, including DD, who at that stage wanted something different. The US breakdown is also interesting. In DDs year eight went to Harvard, which was an impressive high water mark. In 2022 they seem to have got 7 into Stanford.
I suspect ethnicity has quite a lot of influence. As others have suggested London private schools can be very ethnically diverse. This both means more of a tradition of living at home, but also more consideration of international, rather than national, name recognition. Ditto within the state sector, where cost is also a factor. I understand that other than the white middle class, pupils at our nearest and well regarded, state school were often unwilling to consider anything but London options.