Hey! Moving on from a uni rejection can be one of the toughest things that you can go through during school, especially when you've visualised your life at a certain place and have made it your dream for several years. I speak from experience, having gone through a very similar thing. In fact, I didn't even end up going to the country I had initially planned on going to for uni. So you can only imagine the disappointment I felt after years of fixating on this one dream.
At the end, however, I promise you that I wouldn't change a single thing. I'm so happy things didn't go my way and happened the way they did. I've built an amazing life for myself and met people I would not have met. I have accomplished so much more than I would have had I been adamant about sticking to my original vision.
I think you also need to ask yourself why going to Oxbridge, beyond the prestige of those institutions, means so much to you. I understand having visited Cambridge and having fallen in love with the uni, but beyond that, it's very easy to get wrapped up in the facade of what an Oxbridge education means. I know all students do this at some point, but choosing which uni to go to based solely on rankings and prestige is not a great idea. Even worse is to let which uni you go to, and the idea of being an Oxbridge graduate, define your identity. You are so much more than the university that you go to. What will define you in the future is not going to be where you graduate from, but rather what you do after your degree. The way I see it, getting rejected is the best thing that could have happened to you because now, you're actually going to find out who you are, what you like, your aspirations, and so much more.
I don't know what course you are taking, but that should be the most important focus in choosing a programme. Look into the specific modules and teaching structure, and make your decision based on that. I have several friends who have studied at both Oxford and LSE. If I'm being honest, while Oxford was the more 'prestigious' university, it was the LSE kids who actually did better after uni. LSE prepares you very, very well for the 'real world' and that shouldn't be understated.
As for living in London, I wouldn't stress about it too much. I ended up going to uni in London (undergrad and masters), and as someone who is actively not in uni now, I can say confidently that the uni experience in London is very different to what it is like just living in London. Unless you're actively living in the Strand, right next to the LSE campus, your experience will be completely different. London is so big that going to uni in a different area makes it seem like you are in a different world. I've had friends who, just like you, grew up in London and ended up staying here for uni despite their desire to move. They have not regretted it one bit. Just moving into uni halls and avoiding the places they grew up near made such a difference.
It's going to be difficult, but what you really need to do is focus on your A-levels right now. What's paramount is doing the best you can on them. You're going to have so much time after exams to mull over university choices. Just focus on your passion for the subjects you're studying — everything else is secondary!
And remember, life doesn't end after school, it'll only get better regardless of which uni you end up going to.