Hey, so I am coming at this from the perspective of someone who works with a lot of American lawyers. I don't know where you actually may want to practise once you're ready to take that step, but if your interests lie in private practice at an American law firm in the US, I can tell you that the JD is not necessarily that important because you are at somewhat of a disadvantage in comparison to students who did undergraduate degrees in something else and then went to law school for their graduate degrees which is the most common path in the US for lawyers.
Having done 5-6 years of legal study with an English LLB and then an American JD will make you appear somewhat one-dimensional, according to the candid opinions I have heard from several US law firm partners. I know this because it was a path I was considering when I was a law student in England. Make of that anecdotal information what you will.
If you utimately want to practise in the US but are pursuing an English law degree at undergraduate level, I would recommend strongly considering qualifying as an English solicitor or barrister and practising here for a few years. Thereafter, study for and sit the New York Bar and become New York qualified.
While competitive, if you manage to get into an American law firm based in London, once you have that NY qualification, you could, potentially, use that as a springboard. Of course, there are no guarantees, but that is another way you could think about approaching it instead of spending 5+ years in law school and accumulating minimal relevant work experience in the process.