Hi,
So there are actually a number of different ways to qualify in the US.
Firstly, you don't qualify in the US per say, you qualify in a state - most likely international students it will be New York or California.
Secondly, you can do you LLB here and then pursue a recognise LLM in the US before taking a state bar exam. Or you can do your LLB here and then a JD in the US and then take the state bar exam.
It is possible to do courses through providers like BARBRI which prepare you for a state bar exam without having a degree from the US - however, you will likely need a UK common law degree.
There are unis like KCL, LSE, UCL which do joint LLB-JD programmes normally with Columbia / Georgetown but these are highly competitive and are based on your first-year grades. This will, however, put you in good stead to practice in the US.
If you go into corporate la some law firms down the line might sponsor you to take a bar exam if you wanted to relocate once qualified but this is very rare.
Hope this helps!