I was pretty homesick in first year, although I think most of it was just that my halls seemed pretty lonely. I'd been at the same school for more than 12 years and was used to having friends to eat with, meet up with etc. and then going home to my family and it was a massive culture shock to suddenly not have people always around. It took me months and months to make some really good close friends, especially because I'd basically never had to try much to make new friends before... ever!
Anyway, that aside, I basically made myself stay at Uni for three weekends right at the start and felt so miserable and like I was somehow failing at University and letting myself down. Then I caved and just went home for the weekend. After that I went home most or every other weekend, which was perfect for me. I had people around me during the week and then during those long weekends where I couldn't fill all of Saturday and Sunday with social stuff and ended up feeling really isolated, I popped home on the train.
When I moved out in second year to live with some actual flatmates who I knew well, I hardly went home at all because all that went away. So I guess what I'd say is that if it's making you miserable, just go home. Stay if there's something social going on at the weekend, try to arrange stuff to do and don't let home > your social life at Uni in any way, but don't torture yourself. That's basically what I realised I was doing. You'll get used to it and if you view going home as like a safety net, then it'll probably help a lot with how you feel the rest of the time. I felt like a massive failure doing it, but it's better to feel like a bit of a failure than being in emotional turmoil the whole time! I felt unhappy with myself but so so much happier than I'd been before!
I don't know where you lived before but I also think that London is a massive culture shock in itself. Polluted, busy, dirty, noisy... most of the things I love doing are impossible, unpleasant or very limited in London. It's so big and so busy that you never seem to bump into anybody you know by serendipity etc. That's just my opinion. Anyway, stick it out and try to find some sort of compromise where you feel happy is my advice. It will pass at some point, and until then there's no point in leaving a great course where you've made great friends to transfer away.