Just wanted to add my 2 cents, hope it's not bring up a dead thread but I think this one was interesting from my perspective.
I'm just going to work off the original title of this thread "Which UK universities apart from Oxbridge are comparable to the Ivy Leagues?" which is a straightforward question I can answer.
I am currently at Penn which is an Ivy league university (did you know that?) and I'm a full time student from Edinburgh on exchange here. That's a pretty good compare I think. I've also taken a business course @ Harvard.
The first thing is that the educational systems are completely different. The average '3rd year' or 'junior' student in a US university knows a lot more in GENERAL than a 3rd year UK student, while in a specific subject like CS the US student would be laughably behind.
This is a big deal because most other universities in the world follow the UK more specific model, at least in say South Africa, Australia, NZ.
There is also a fundamentally different culture. Remember Ivy League universities are private, none of the UK ones we are talking about are. This is a big deal. The reason why US universities are more 'heard of' is that people are in general more proud of them, former alumni more often will name drop and promote their previous schools because there is more of a culture of a life long attachment because these US universities are seen as part of someone's identity. It is NOT just a place of academic learning. Many people come to these Ivy Schools just because they are Ivy schools and it's about the brand, prestige etc and they spend the rest of their lives going on about it. Seriously. People generally don't go on about it as much in the UK, because it's not seen as this huge attachment to your ego.
Anyway, simply put, the academic standard of these Ivy Leagues is a bit of a joke. It is no where as near intense as Edinburgh for example, because they must cater to a general audience of students. Yes, I'm taking an undergrad level course with PHDs, masters students in different fields etc. There is little structure. Some may like it, some may not but there are definitely very different structures and a UK university will be much more field intensive and specific. This university does not cover anywhere near the depth of an Edinburgh CS degree.
It's also quite easy to get an A. There are open book tests and many team projects.
So do not for a second think that UK universities are not comparable to ivy leagues. I'd say all of the Russell Groups + more. In terms of prestige, many professors at Ivy League universities work with or have even studied at the Russell Group. Many of the students have heard of tons and studied at or applied to many. Especially Imperial, UCL, Oxbridge, Edinburgh, Warwick, Durn, St Andrews, LSE.
If you are an undergrad I strongly recommend studying in the UK, you will get a much more solid foundation in a subject than any Ivy in the US. If you want to meet people at a huge expense and harp on about it for the rest of your life, go to an Ivy.
NB: Also, in terms of selectivity you forget a VERY important point. ANYONE can apply to an Ivy League university. EVERY application is considered. Yes, you can apply if you 'A levels' sucked and many many many people apply and get rejected just 'to try'. People in the UK do not all apply to Oxbridge if they know they have no chance. People in the US definitely apply to Ivy's just because they are Ivy's and it would be amazing if they could actually get in. There is no minimum requirements for universities like there is in the UK! % Rate acceptance numbers are hugely misleading. The UK is far more selective as it is based purely on academics. You can get immediate access to an Ivy with bad marks as a 'legacy' student (ie your family went to the university before). I know some of these students and they are less than stellar. You will not find that at Edinburgh. US universities are more about if they want you around, UK have strict policies that they follow with no flexibility.
Edit: I also want to emphasise not to rely too much on university rankings. It's like ranking countries or even ranking people. A university is a big place and its really hard to quantify everything down to a number. Department specific information is way more valuable. For example, Harvard is NOT the 'top dog' university. I know at least 4 students who turned down Harvard for Penn because a specific department at Penn is better (ie the Wharton School is considered the best business school in the world). Again, MIT CS is very strong, most would choose over Harvard. This is a more reasonable way of looking at things and easily confirmed by talking to students in the US. This isn't *so* apparent in the UK but it's there.
By the way, the Ivy League universities are modelled off Oxbridge to an extent (Colleges, traditional, architecture etc). You know why Oxbridge has such a huge rep? Because they also try to more follow an Ivy pattern of creating a sense of community that students are more likely to self identify with for their entire life. No other university in the UK does this because they fail to implement a similar style College model which is present at Ivy's through the informal links with professors, alumni etc. Just today I had lunch with three top professors from Penn, that would never happen at Edinburgh - it happens at Cambridge though. This is a bigger deal than you think.
Edit Edit: I also just wanted to add I also studied in South Africa for my first year at a really bad universities you've never heard of. It's not ranked in the top 1000. It stands up shockingly well to Edinburgh and Penn across the board. My personal advice is that comparing universities is really like comparing countries. There are just too many variables and personal differences. As long as you go to a university with a computer lab, library, professors and the base departments don't for a second think that Harvard or MIT is some magical kingdom that's a million times better. It's easy to think that. But it's just not true. Get your self confidence from somewhere else and don't let university rankings ever define you, your knowledge or your university!
Edit Edit Edit: Also want to add I studied at Keele University, not a high ranked English university. Guess what? The CS assignments for 1st and 2nd year are the same assignments at Penn, Edinburgh and Caltech. Standard exercises from standard textbooks. Higher ranked universities do not have magically superior courses! Professors would rather be doing something else, everywhere! The only exception is postgrad - go to a good lab where they're doing specific research in what you're interested in! When you're doing this, you'll see the last thing you'll look at are rankings!