I can't copy pages out of the FAA Flights Standards Information System on the iPad but in the section giving guidance on how to process 61.75 applications is a paragraph on explaining the regulations with the new holder of a 61.75. It says to tell them they need a FR before using their FAA cert and that it must be from a FAA CFI. Go about half way down the page to 5-603 I.
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While your UK medical is fine, I'd say the flight school doesn't have a clue.