Getting into a top uni (UCL, Imperial, etc.) with a 2.2?

Well you were talking about global reputation, so I don't need to go any further than seeing if a university is known by different groups of people in different places, and if it is talked about.

Most people overseas know only 'University of London'.

LSE probably has the best international reputation out of LSE/Imperial/UCL, at least among non-academics. I would expect more people in the US to have heard of it than the University of London, particularly in the banking/finance industry (which is where most LSE grads want to end up anyway :s)

Its actually 10% and it does seem to matter a fair bit, although obviously nowhere near as much as The_Right thinks. If you look at the invididual QS component scores then Imperial is 14th on academic reputation while UCL is 16th. Similarly on Employer Reputation, Imperial is 9th and UCL is 22nd. Both are more accurate reflections of those universities than the (slightly absurd) overall scores that put them in the top 5.

The reason why it matters so much despite 'only' being 10% is because the gap between the UK and US universities is much, much larger on those scores than it is on the (more important) reputation scores. For example, on international faculty Imperial and UCL get 100% and 97% respectively, while Stanford only gets 75% and Princeton gets 76%. Similarly on International students, both UCL and Imperial get 100% while Stanfrord gets 76% and Princeton gets 69%. Wheras if you look at academic reputation, both Stanford and Princeton get 100% while UCL/Imperial get 99.9%. Similarly on most other scores, the difference between Stanfrod/Princeton and UCL/imperial is only 1-2% at most. So when you have that enormous 20%+ gap on the international scores, it has a huge effect on the overall weighted ranking despite its low weight, which results in UCL/Imperial finishing above Princeton/Stanford (and others) despite clearly not being quite on that level (although both are very good universities)

Again though, this affects all UK universities equally, they are all overrated on QS. ARWU is probably the most accurate ranking.

It's just how I view things, and how many others do too. Generally speaking (and ignoring isolated examples of grade inflation), its harder to get high grades at top universities because you are competing with better students, and both exam difficulty and grade boundaries are partially set based on the distribution of student ability. You can't just say "a 2.1 is a 2.1 regardless of where it comes from" and that isn't how anyone views it, neither in academia nor industry.

You will be on safer grounds with a 2.1 from a decent Russell Group than a 2.2 from Oxbridge/etc for sure, not least because you won't be automatically screened out at places that use a very inflexible set of criteria and simply refuse to consider anyone with a 2.2 (more common at oversubscribed industry grad schemes than postgrad admissions). But if your 2.2 is from a top place and you can tell a good story about how you expect to perform better in the future, there is a decent chance that you will be able to sneak in even if a 2.1 is generally preferred.

Be very careful with assumptions like these.

Maths is always the example that is brought up, and it may well stand given the structure of Mathematics degrees that the difficulty level varies considerably from institution to institution. I'm sure there are other subjects that are similar to this.

However, this does not hold true in whole swathes of other subjects, in fact I would argue in the majority of fields. In the Humanities, for instance, work is often externally moderated or marked externally exclusively, to ensure standards are kept uniform across institutions, and likewise considerable effort goes into keeping the demands of programs equitable.

I'd also caution against your willingness to devalue diversity of nationality as a criterion for judging universities. Try majoring in Philosophy at a culturally homogeneous midwestern US institution and you'll understand why it is important.

Also, UK universities operate on a different financial model to those in the US. endowment isn't a good basis for comparison.

Yes it can be done, one of my good friends just got accepted into an MSc at both Imperial and UCL despite having a 2:2. Also a few months ago I was giving advice to someone on this forum about their chances of getting onto a different applied mathematics) MSc at a top London uni with a 2.2. I told them just to apply because I thought they had a decent shot, and they also ended up getting accepted.

The quality of your undergrad institution will matter. Getting a 2:2 from Cambridge is not the same as getting a 2:2 from Manchester, which is not the same as getting a 2:2 from London Met. To be frank, a 2:2 from Cambridge isn't that much worse than a 2:1 at a low/mid-tier Russell Group and this will be taken into account (both the people I mentioned above got their 2.2s from top universities). Unless you are going for one of the ultra-competitive Masters programs, you probably have a decent shot, although obviously don't get your hopes up.

The personal statement will also matter, as others have said.