You appear to be asking if you can use a FAA commercial certificate to fly for compensation or hire in the UK in UK registered aircraft. that is a question to be ansered by the CAA...becuase you're talking about operating in the UK.
You will not find answers to that question in the AIM, as the AIM is not regulatory. It's a guide compiling information from various regulations, advisory circulars, etc, and serves as a standardization document detailing particulars of airspace, procedures, roles, etc.
You won't find that question in the United States Code of Federal Regulations, Title 14 (sometimes referred to as the "FAR's"), either.
You won't need a first class medical to obtain, or exercise the privileges of a FAA commercial pilot certificate, in the United States. How you use it and what's required in the UK, will be a local issue with which the FAA does not concern itself.
When you do use the FAA certificate, you are subject to the regulations set forth by the FAA, and to the regulations set forth by the country in which you are flying. The FAA does regulate and prescribe what can be done for hire when operating under an FAA certificate. It doesn't tell you what registrations you can fly, however.