How many years have you been in college? That does tend to complicate things a bit.
If you’re going in as a first year, your ACT and SAT are both a bit low for Oxford and LSE. Additionally, while it’s no longer a complete, set in stone, requirement, there is a bias in favor of UCAS Group A courses when compared to UCAS Group B (Group A courses are those which, in the US, would grant a full year of credit, while Group B offers only a semester of credit).
You mention that you’re most interested in Big Law in the City; as others have mentioned, your training as a solicitor would all but max out your visa, but you are right to point to the visas given by the bigger firms. However, those visas are dependent on proving you’re invaluable; a solicitor trainee is not always such.
Of course, you can return to the US and practice law, but you’d likely need an LL.M for that. CA, NY, and TX, among others, permit this. Instead, I’d consider an accelerated JD as a way to make sure that you can practice in every state.