I doubt if Grey Gardens will ever play the West End. There's just no market there for anything that's not a revival or based on a well-known movie.
Musicals currently in the West End: Chicago, Les Miz, Phantom, Cabaret, Dirty Dancing, Lord of the Rings, Billy Eliot, Wicked (also rumored to be in trouble) and, for a limited run, Kismet! Plus Joseph and the Technicolor Nightmare and some jukebox junk. Not an appetizing menu to say the least.
Faced with these Loreleis, what man can moralize!