FWIW, I seem to recall that very few (if any) UK schools would allow a new customer to self-fly hire one of their machines without some sort of check ride, even with a JAA PPL(H). The general consensus was that a new customer who hadn't flown in the country before (ie had done all of their training in the US, even if it was to JAA standards) would require more 'familiarisation training' than someone who had trained in the UK.
This is true for schools in the UK and USA. Any self-fly hirer is going to want to see a checkout of the person doing the hiring before letting them head off alone.
US can be cheaper and more fun, but not everyone can take the time, or get the money up front to do that. I will concur though, that HAI/Bristow Academy was a good place to learn the basics, and at that time anyway, instilled a good safety oriented approach to ab initios.
Good Luck OP!
BG.