Hi, I took the LSATs twice in London (the second time after the TestMaster's weekend course...if they have it at LSE's Rosebery Hall again, please note that that's a dorm in Islington and not on the campus on Aldwych) and coordinated my second application cycle (this one) from South Africa.
I would suggest giving your US address for LSAT registration, so you'll get fee waivers. However, take full advantage of the "address valid until" slot on the applications. I notified all the schools to use my (US) permanent address when I came back from South Africa, but at least one school sent my acceptance there.
I'd really advise starting at least a month and a half before you want to have your applications submitted. Processing my foreign transcript, in particular, took ages, and I was ready to submit my applications fully two months before I had all the transcripts and recommendations in. Just know that it takes a lot of time, but thank goodness for the internet!
Let me know if I can help you at all! Enjoy London! It's the best city in the world and I miss it every day....