Focus on applying to law schools first, and their international office will be able to help you out with your visa and, (if they're approved by the US Dept of Education) should be able to help with any US-Student loans you take out (after filing the FAFSA, of course).
Two things I would advise caution on:
1- It's going to be quite expensive. Especially if you attend a University in London. Since it would be an undergraduate course here, would you be able to afford it for the full length of the course? With US loans, you have a maximum of either $31,000 (if anyone can claim you as a dependant), or $57,000 (if you file as independent). With each of those, only $23,000 is subsidized, meaning that the moment any unsubsidized loans are dispensed, they start piling on interest....which really has a way of adding up. (Especially when it becomes capitalized..)
2- Do you intend to practice law in the UK or US?
If in the UK- you need to consider whether or not you will be able to stay once you finish your course. The Post-Study Work Visa route is now closed, (however there are talks of reopening it for "Entrepreneurial students" or something like that...but I don't know when that's going to happen.)
Or, if you have to return to the US--Will your UK Law degree mean anything there?
(I did just find a joint LLB/JD Course at King's, which would probably work out well, if you didn't mind going to Columbia University for two years..but I'm not sure how common this sort of thing is)
The first thing to do would be to research your options. Then consider what's best overall. Good luck with your decision. Also, if you do end up going through the UK Tier 4 Student Visa process, there are a few threads from previous years that are quite helpful, so I suggest going through those to get an idea of how things usually play out.