I noticed the nearest AME to me is 'FAA approved' according to the details on the CAA website, so I guess I could get both an FAA and CAA issued at the same time?
This probably sounds a daft question, but I presume it involves one medical but two sets of paperwork/fees?
I'm guessing by your location you're thinking of Doc Ramsay in Cheltenham.
Get there early if so, all the forms will be ready for you, and he'll do both for a very reasonable fee. If you're thinking of maybe a standalone FAA in the future then that might be the way to go (And in fact it does no harm getting both done at the same time anyway).