Please suggest which one you would choose and why

Yale by far... I do not know where do you live, but I think that in the law industry the prestige of US Universities is bigger than UK in the majority of countries.

Also you are not talking about any school of US, you are talkning of Yale! the number one law schools in US.

I suppose you would like to enter into academia, then I would suggest you choose Oxford to pursue PhD. PhD degree is recognized wordwide, while SJD is not. Oxford own bigger names than Yale in some countries (not sure where you are from).

[quote] I suppose you would like to enter into academia, then I would suggest you choose Oxford to pursue PhD. PhD degree is recognized wordwide, while SJD is not. Oxford own bigger names than Yale in some countries (not sure where you are from).[/quote]

If you are distinguishing between degree names, then you should know Oxford offers the DPhil, not a PhD. The OP asked about LLMs though. One can just as easily attend the Yale LLM and then apply to the Oxford DPhil. In fact, the BCL really isn't considered preparation for the DPhil as Oxford still expects you to either do an MPhil (most common route) or probationary year before properly entering the DPhil.