Original post by avarI did quite extensive research on both the US and UK educational systems last year when I was applying. I've also got a ton of anecdotal and empirical evidence both from my own experiences and those of friends.
I was in a very similar position to you last year. A rejection from my first choice Harvard meant I had to choose between offers from the LSE (Government and Economics), UCL (History, Politics and Economics), Columbia, Berkeley, Dartmouth and Cornell - I ended up choosing the LSE, with Columbia as my backup.
I'm now 3/4 into my first year at LSE and I've honestly loved every minute of it - I haven't had a single regret about turning down Columbia. I've found the atmosphere at the LSE to be intellectually nurturing, without being stifling or prescriptive. Socially, I've found that you aren't at all limited by the lack of a campus vibe at school - London really is your oyster. As a student in a major city you meet a huge number of new people on almost a daily basis. If you're outgoing and engaging enough you'll make some cool friends who do cool things - I've been to art galleries, underground clubs and even into the House of Commons through people I've met in and around Uni. Being at an Elite uni in such a big city really, really opens a whole lot of doors - that should be a huge factor when making your decision.
I will say however that the careerist *springweeksareallthatmatters* mentality does exist at the LSE, and it is shoved in your face at freshers fair. However, if you look for it, there is an abundance of awesome stuff to do in and around campus. There are a vast array of societies, and clubs, and a whole load of interesting, international people. This is a critical difference for me. Elite US schools typically have a 10-15% international population - the overwhelming majority of the people you deal with in the States will be American - for me this is a huge negative. At the LSE (50% international) I've had the opportunity to interact with bright, driven kids from all around the world, each of whom bring a different perspective and world view - for me this has been invaluable, and probably the highlight of uni so far.
I personally find that British uni life is on average a lot more mature than American life. A lot of the typical 'college experience' I find contrived, and forced. I personally like that LSE students are kind of left on their own when it comes to socialising, it allows you to find your own friend circle and scene.
In terms of education; there are VAST differences between the type of education you receive at American and British Unis. One is not normatively better than the other, they just suit different types of people.
The American system of education is based around year-long effort, and is highly structured. Each of your classes will give you assignments which you have to do weekly, each of these will be marked by your teacher, and contribute to your GPA - this is effectively your degree classification. This means you really have to be on top of your stuff week in week out if you want a 3.7+ (1st class equivalent). However, on the flipside, this also means there is less riding on final exams. Personally, I find this system a little babyish - I much prefer exam based testing.
The British system of education (what you'll receive at the LSE and UCL) is far less prescriptive than the American system. From what I have observed, the model is lectures give you the information, classes allow for discussion and critical engagement, and then you are examined at the end (although LSE now have Jan exams fro maths and econ courses). There is very little 'structure' to the course. This means that the Uni won't tell you when, where or how to study, they essentially leave it up to you. This suits me perfectly - I can miss deadlines and skip classes during the year because I'm busy working on my startup, playing sports or just going out, and still get a 1st overall.
I terms of the structure this comes down to a personal decision - exams vs year long assessment.
Additionally, possibly the most key aspect of the educational differences is liberal arts vs specialised studying. At the LSE, you study mainly what is written on your course code (you can take an average of an outside option a year from any department). In the states, you kind of 'shop' around for a while. There are advantages and disadvantages to this again. If you don't know what you want to do, you have leeway in the states. If I wanted to switch from my LSE course to something dissimilar like Physics, I would have to change unis and start again. In the states you typically have until year 2 to declare your major. However if you are set on the general field, at LSE it is quite easy to switch degrees into related courses. For example with maths and econ you can pretty much switch into any department in second year except for possibly law.
In terms of content, I can describe the difference between the US and LSE as you get a lot more easy work in the states. LSE doesn't really give you mountains to do, but the work is far more advanced than what my friends in freshman year of elite US schools are doing. I know a Pol Sci major at Northwestern in his second year who has almost an identical reading list to my introductory course on political theory. However, you have to trade depth for breadth when comparing the US and UK.
There is one caveat however. If you plan to do any science/engineering go to the US. If you think there is at all a possibility that you may do practical research, and require access to high tech lab equipment, the UK schools cannot compete with the US in terms of funding. If your interests are more theoretical, then the above point is moot - as there are plenty of opportunities for original undergraduate research in both systems.
In terms of exit opportunities, it really is a wash. Neither LSE nor the US unis you mentioned have the historical 'old money' prestige of an Oxford or a Harvard, however, they are all universally known as top tier. This means that if you can't get a job a Goldman Sachs/Mckinsey/Clifford Chance or wherever else you may desire out of uni, it is completely your fault. Obviously, where you go to school will likely dictate where you start working, so geography may be a factor.
In conclusion, all the universities you've mentioned are elite and world-renowned (UCL is probably a notch below the others), you'll likely be fine and enjoy yourself and learn a lot wherever you go. However, this really should come down to what kind of person you are, and which system suits you best.