Gordy. Thanks for this...but I'm not sure that this answers my question. The section you highlight concerns flying 'aircraft of U.S. registry' - I'm talking about flying any aircraft (i.e. in effect G or N reg) in the U.K.
In addition, I think that 'limitations and restrictions' (which are carried across) are different to 'ratings and privileges' as defined on the EASA license. I cannot see any specific mention of 'ratings and privileges' being automatically carried across from the foreign (i.e. UK) license to the FAA certificate. Just above the para you highlighted it says that "(the pilot) ..is limited to the privileges placed on the certificate by the Administrator" - but that's not quite the same thing.
It does seem unlikely to me that this is the case - but the reason I ask is that I am assured that there a significant number of pilots operating in this way - flying helicopters on their FAA certificates for which they are not specifically rated on their EASA licenses - in the UK currently.