Seeking US employment with a British doctorate might create problems in a social ways, however.
1) Make yourself aware of the US reputation of whatever British university you are thinking to attend. The 'prestige' of the school where you do your doctorate will count, at least in opening doors.
2) Consider the intellectual cast of the British school you want to attend and the US schools you might later want to work at. Hiring institutions will want you to bring something different, to fill some niche not filled by current faculty, but they usually won't want it to be 'too' different from the existing tenor.
3) Make sure that your British doctoral supervisor is well connected in the US. Getting a leg up in the academic trade is less a matter of what you know or even who you know, but of who your mentor knows.
4) Make sure you really understand the entrance requirements for British PhD programs. In the States, at least in my day, one typically completed a masters degree in the course of a doctoral program (which your department might make terminal if they didn't think you were well enough qualified to continue on). It was in the course of that master's work that one developed a doctoral level research hypothesis and defined the literature relevant to investigating it. When I worked in Britain 30 years later, one was expected to have a developed research hypothesis and some actual or, at least, likely sources of financial support to be admitted to the PhD in the first place.
5) Be very clear about where and why you are applying to a particular British doctoral program. The UK clearly intends that promoting British degrees to overseas students be a money-making proposition. Be sure you get good value for your money, even if the money comes from a third party funder.
Ask these questions of as many professors and students as you can access.