Actually, the IStructE is only for structural engineers, and it's the ICE that governs civil engineers. The UK chartership program is very cumbersome, involving mounds of paperwork accumulated into a portfolio and signed by a PEng a multitiude of times. There are experience and continuing education requirements, and thankfully most UK firms have graduate training programs to lead you through the process.
If you're trying to get chartered through the ICE in the shortest amount of time, then you need to work for one UK consulting firm for the full period. It'll be the quickest and easiest way to chartership, but you'll compromise on flexibility and ultimately on salary. The larger UK companies are multinational and I was able to travel and work overseas twice in two years and was slated to go off to work in Australia for a year when I quit.
The process in the US is much simpler, but has its own pitfalls for those with foreign degrees. There are two exams, the Fundamentals of Engineering and the Principles and Practise of Engineering, as well as education requirements. If your UK engineering degree is a 3-year degree, you won't qualify for registration in the US. (In fact, the ICE is tending toward a masters degree as the minimum education requirement anyway, bringing you up to 4-years in university either way.) See
the FAQ on the FE for more information.
While the ICE, in principle, will recognise your experience in the US, you will have trouble with getting someone they recognise to sign you off, and you may have trouble getting the diversity of experience that they require. On the other hand, they more than likely will recognise a PE, so if you're able to, spending four years in the US and getting a PE may be the quickest and easiest path to chartership in the UK. I'm not sure if the
Washington Accord covers chartership or if it's just limited to education, and remember that many states don't recognise it (but hopefully that will change one day).