Being fellow NATO members with Greece didn't stop Turkey from invading Cyprus to prevent Enosis. Sure, they might give more pause, but what stake does the US, UK and France have in Cyprus that is more important than locking the Soviet Black Sea fleet behind the Bosporus?
Well the UK has a lease/rental agreement for RAF Akrotitr, and US aircraft use the base so I don't see the point in Cyprus becoming part of NATO anyway.
Also the said agreement between the UK and Cyprus is very resentful for a good percentage of the population, if the Cypriot govt made moves to become part of NATO I could see a situation where the govt falls and replaced by a staunchly neutral one and an end to agreement and the forced closure of the RAF facility.