What I mean by ‘Universal’ is to be used anywhere.
is this the case or do pilots flying between America and the UK just have both CAA and FAA ATPLs?
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If you are flying a "civil aircraft of the United States" (an "N" numbered aircraft) you may operate that aircraft between America and the UK (per your question) with just an FAA issued pilot certificate. See 14 CFR 61.3 Requirement for certificates, ratings and authorizations.
A separate pilot's license issued by the CAA when operating an "N" numbered aircraft in the UK is not required.
Annex 1 of the International Civil Aviation Organization (ICAO) creates mimimum standards for crew licensing that all ICAO member nations recognize. The ICAO is an agency of the United Nations, and was established by the Chicago Convention on International Civil Aviation in 1947. So far, every UN member with the exception of Lichtenstein and the Holy See (no airports) have ratified the Chicago Convention and are members of the ICAO.
So yes, an FAA ATP license is recognized by treaty in nearly every UN member state.