I am currently in a similar dilemma myself (although am yet to receive an offer from LSE- so this may a be a purely academic exercise).
However, LSE to me very much seems to have an international law bent. There are modules on BITs, International Trade even International Tax (I seem to remember). UCL is slightly more domestic and traditional in what it teaches. Maybe this is a reflection of their relative demographics? I personally am more of a domestic lawyer, so at the moment am veering towards UCL. Certainly I have no great love of international law.
In terms of reputation domestically- I think there's very little to choose, they are both excellent law departments. Possibly more importantly, in London you're in the thick of it- you've got the Inns of Courts, the Chambers and everything else for barristers- and of course the law firms in the City for solicitors. Both universities will have excellent links with these institutions. It's certainly a very difficult choice!
All in all go with your interests- it will be an intense year, and there are significant differences in course content. I hope this helps.