My elder brother studies at manchester and I am going to (grades pending) LSE next year and I have a number of friends at LSE so maybe i can help you weigh up a little better...
Manchester = a student city, always parties going on and always things to do. however being 1 of like 50 or 60,000 students makes you a small fish in a big pond and in many ways it can be much more rough than London as being in central means that its pretty much always busy and always well lit.
LSE = definitely less sociabale than manchester however realistically it seems to be that it is what you make of it and if you are determined to, its not hard to have a good time. t say that noone goes out would be false because the people that I know there certainly do..and they are very very active citizens meaning there is a tonne of stuff you can get involved with. sure manchester has more volume wise, but you cant take on a million things anyway so the choice in london should do you just fine.
Manchester again being student orientated is very cheap - my brother pays £77 for a two bed flat (so £77 from his flatmate too) which is pretty nice and in a decent area. Also the clubs and stuff again are cheaper I would presume
LSE - we all know london is much more expensive however you can get halls from as little as £90 and from everyone who goes there, if you stay economical then its not so hard to live. there are plenty of part time work opportunities due to the tourism industry and london pays fairly well. Also as mentioned, you get a much larger student loan (upto £6200 which is over a grand more i think) and on top of this, if you come from a low income background LSE offers a bursary as well as a large number of scolarships which can significantly ease the financial burden. So yes, it costs more, but there are plenty of ways in which you can cope.
Manchester: its got a huge huge student body and although you rarely heaer about its bad teaching, my brother often complains that it isnt great and considering the volume of students he often has problems with really bad administration at the uni that can really piss him off.
LSE: yes it has a rep for bad teaching, but as far as I can tell most people get along fine despite it. Manchester has also got an international teaching staff so the whole english problem may well apply there and in addition, LSEs teaching prepares you to be much more of an independent learner which can only be a good thing.
Manchester is certainly making inroads with its huge spending plan, and they reckon they will get into the top 25 globally within something like 5 years. whether or not that happens i dont know. howeveer LSE has an enviable reputation as a world leading institution and although it dropped to 59th in the world this year, this was due to a change in the make up of the tables that favours the natural sciences and disfavours the social sciences..tehrefore dont pay much attrention to that.
I can see where you are coming from in wanting a studenty lifestyle and therefore the appeal of manchester and newcastle (sorry i havent commented on there but i know nothing about it!). i suspect that my brother does have a more studenty lifestlye than I will when i go to university, however that said i know people at LSE who seem to go out just as often as he does, maybe once a week less at max. therefore as i say, dont listen to people who say LSE has no social life - it will have one so long as you want to find it. although you dont want to become an investment banker, not knowing what you want to do would say to me you should pick the instiution who's name carries a much better name, incase you do want to go into some uber-competitve field.
if you want to go to LSE you can have the fun three years that you crave. with so many people wanting to go there I would suggest that reputation should preceed cost (which is manageable) and that as long as you want fun you will find it.
wow that was a long post, i bloody hope it helped lol
good luck choosing!