I did my FAA CPL/IR four years ago at Crystal Aero Group, Crystal River, Florida. They have both 61 and 141 approvals. I was very happy with the experience and would happily go back, but they are only approved for FAA licences - although they are also specialists in training non US Citizens.
I'm not aware of anywhere training USA and UK or EASA licences together. My advice (as somebody who holds all three) would be to do the EASA or UK licences first (or both together, they basically have the same syllabus, and you can do groundschool, flying course and skill tests together, just you'll need to do two sets of exams. THEN take the FAA CPL and IR somewhere. The FAA groundschool is massively less than in Europe, and apart from air law, you basically won't have anything new to learn. The American CPL syllabus will take a little new learning, maybe half a dozen hours for some manoeuvres we don't have any use for in Europe - the IRs are basically the same in content and difficulty, with the minor additions in the USA of having to fly a partial panel approach, and circling approach, neither of which will challenge anybody who's already learned the UK or European IR.
At risk of stating the very obvious - the UK is in neither the EU nor EASA any more. A UK class 1 is useless for getting an EASA licence (and vice-versa) so if you want to do both UK and EASA licences, you'll need both medicals. Finding an AME who can do both together isn't hard. UK citizens need a work permit to work in the EU (except for Ireland), and EU citizens need a work permit to work in the UK (unless they are Irish).
G