Gosh what great offers!
This isn't my field of study so the courses themselves I can't comment on and it sounds as if you have some particular interests of your own that will guide you there.
To be honest they are all very good universities. LSE is obviously very well thought of (though the course sounds a bit eurocentric)
In terms of campus life and areas of the country:-
Birmingham... lovely campus, Russell group uni. redbrick, BIG city, buzzy, multi-cultural, quality museums and theatres, art galleries, tradition of civic pride, all the usual stuff associated with big cities such as some crime, poverty etc. University in nice area of city. Brilliant transport links to the North and South. Hour and a half to London. Excellent internal travel... buses and small railway lines etc. Great night life. Of interest to you might be the fact it has an international conference centre in addition to the National Exhibition Centre up the road.
Sussex... campus on the downs, close to Brighton/London... alternative scene, Brighton known for its LGBT community and voting in the first 'Green' member of parliament, great for students, arty, trendy, 'London on sea' some people call it.
Exeter... lovely campus, wins accolades. Small county city serving rural area. Can get to coast and Bristol in half hour. London within two hours or so. Very relaxed, very friendly. Zooming up the tables because of big investment in facilties and courses, forward looking. South West primarily white, rural, relaxed, slow paced, pretty. Warmer in winter.
York.. traditional, attractive city with well respected university.
Sheffield... old industrial city, third biggest in UK. City university rather than campus based.
Warwick... respected university but really it is in a field outside Coventry which isn't the most desirable location in the world. Can reach Birmingham though.
Royal Holloway.. in very, very rich part of country... cut off, on top of hill just outside a small commuter town, journey to London slow, transport poor (better to have car) conservative students on the whole, can be stuffy. I went there and hated it as its a bit insular. Lots of people love it though.
Bath... very well respected university indeed. Beautiful Georgian city on tourist route. Very Jane Austen! 15 minutes from Bristol.
Liverpool... buzzy, bouncy, friendly city with tradition of socialism. All the usual stuff about cities, crime, poverty etc in parts.
Ediinburgh... cold!!!! Elegant, grey capital city.
Sorry to be vague and I'm sure you know most of this... but just in case.
Good luck choosing.