I don't know how likely it'd be. The British were in Greece until 1940 or 41, I believe, then they evacuated out. But they definitely maintained close ties with some of the Greek groups.
However, just for the sake of argument, had the Soviet Union managed liberate Greece after WWII, then I can see that it would be very likely that the northern end of the Bosporus (Turkey had both ends under it's territorial control) would probably be more heavily militarized, even to the point of the US sponsoring a Keyseri-level NATO base there.
Also, in my opinion willing to bet the Turkish invasion of Cyprus wouldn't have happened at all. The US and the Soviet Union would probably have clamped down hard and made sure that, even though it was bound to make for some emotional, shoe-slamming speeches, neither the US nor the USSR would want 2 of their Allies to get into a shooting war in the Med.