Welcome to the forum.
The licensing isn't as easy as you may think. The USA and U.K. Run on completely different licenses as you will have found out with your research. It's all about being efficient with your decisions and money. The syllabus for both licenses are vaguely similar but there are quite a few stark differences. Hence why it takes quite a lot of time, effort and money to fly in the other licensing authority's airspace with the other license.
My advice would be to research which schools do both FAA & EASA. They should have a training programme set up so that you can train for both licenses and sit separate flight tests at the end.
LZ