I seriously doubt N-registered aircraft flying abroad have increased in number.
ChickenHouse, I don't think that is correct. When I was hanging around airfields as a spotter in the early 60s and then as a PPL later in the decade, I don't remember seeing or even hearing of any N-registered light aircraft, with one exception. There was a Beaver at Booker that was used by a flying club that catered for USAF members stationed at RAF High Wycombe.
I would be interested to know how the UK-based N-reg fleet came about. Was there a change in legislation or was it a practice that just grew as more and more pilots realized the advantages?