No clue, but aside from the hyper-flaming types, the more reserved, but still classic stereotypes have a very northeastern USA sound to them. I think we’ve all probably met a few perfectly straight guys that have that upper-crust accent that makes “gay” or at least “momma’s boy” ring loud.
My horribly imperfect hypothesis is that at a critical juncture, namely the rise of modern media (TV and radio), many gays had already been well established themselves in artistic fields. As their favorite clientele were such upper-crust ultra wealthy folks with money to burn, to fit in more comfortably and work more profitably, they either purposely, or subconsciously adopted a similar tone. It puts the paying customers at ease if you sound and act like them, and if you’re working in the arts, you’re not dealing with too many blue collar customers.
And, even if you aren’t in that field, if your boyfriend and other gay friends are, you’re bound to adopt similar tonalities, because you’re around it all the time, and also, it puts you at ease. Someone that speaks that way regularly will pick out an “impostor” quickly, so there’s a possible safety factor at work as well (or for some, a “newbie” factor). If that’s your personal life, eventually it’s going to peek through, no matter how hard you try to hide it.
Of course, tones today are more a factor of mass media, and there’s not as much secrecy involved, so it’s probably dying a rapid death, aside from those who enjoy flaunting the difference. For the same reason, those quirks are now ingrained in the world’s mind, so any poor soul who naturally adopted his mom’s wealthy eastern accent gets pegged as gay.