It is not ATC job to query pilot or aircraft paperwork. They have to assume you are legit.
In some cases they can "obviously" see the one or the other isn't legit, and they might give you subtle hints to not proceed
I know of a case where an IMCR pilot tried to depart IFR from a French airport. He filed an IFR flight plan but (as one would expect, without knowledge of such) this didn't validate via Eurocontrol.
The ATCO just said "the flight plan is not a valid IFR flight plan" and absolutely refused to be drawn on any details. It was his way of saying "look I know you haven't got a real IR"
Anyway, from the above, it sounds like an ILS logged to Jersey by a UK IMCR pilot would be legit - in the sense that nobody would be able to query it afterwards, especially after the voice and radar tapes have gone. You could not say the same for an ILS into Heathrow which is 100% Class A and such a logbook entry would be obviously illegal even if seen 10 years later. I would never even dream of logging something that would be self evidently illegal when examined at some later date, by someone who can see what paperwork I had and when I had it.