So far as relationships go, there are advantages and disadvantages to living in London.
London is, relatively speaking, not a very social part of the UK. People who live in London aren't as sociable as those living in smaller cities and closer to the countryside. Generally speaking, you will find that people outside of london are a lot more friendly and engaging. You can sit on a train in Manchester or Nottingham and have random people come sit next to you and start a conversation with you. Strangers say Hi and people like socialising. This does not happen in London where everybody tends to stay away from everybody else. Strangers will not start conversations with you and if you stare at the wrong person you might just get beaten up (that is an exaggeration so don't take it seriously). Also like you said, London is a lot more expensive than Manchester and Nottingham. That does make socialising a little more difficult for people, particularly students. The Kings College campuses are all very depressing. I am only basing this on my very superficial visits to the university so don't take my word for it. All of UCL's campuses, with the exception of the School of Pharmacy, which is situated in the worst place in the world, are amazing and in the middle of the hustle and bustle of Central London.
The good things about London as compared to Nottingham and Manchester is that it is an extremely crowded city with an extremely diverse population so you will never be wanting for friends. Another good thing about London is that getting around this city is very very very easy. The transport services are very reliable and very frequent and very timely. So you can get around easily as compared to the north of the UK where transport services in the majority of cities are privatised, slow, dirty, difficult and often an untimely mess. There aren't many Finnish people living in the UK but there's a greater likelihood of finding nordic people in London than anywhere else in the UK. There are a ton of universities in london, each with tens of campuses around the city and and so London has hundreds of thousands of students so the student scene is HUGE as compared to other parts of the UK, say Nottingham or Manchester where there are only a couple of main universities and little to no diversity.
Life, though expensive, is easy in London as compared to other cities in the UK. London has anything and everything you could possibly think of or want 24 hours a day, 7 days a week, 365 days a year. Life is a lot more difficult outside London where services aren't available to you round the clock. That makes socialising considerably more difficult for people outside London.
If you become a student at one of the many "University of London" universities (Kings, UCL, St Georges, LSE, LBS, Goldsmiths, City etc. etc.) you gain access to the other universities that are part of the federal university and some of their services. That, of course, gives you access to their students and their campuses. I am a student at UCL. I often visit friends at King's College and Queen Mary. My UCL student ID card allows me (limited) access to those universities. This allows you to form relationships with students in other universities.You don't have this option outside London.