This is a guess, but I'd say the answer is actually in the question:
"Accents are usually geographical, not cultural."
Accents aren't 'geographical' - where one is in the world doesn't (entirely) create and influence an accent (it has influence, sounds move differently in different environs, etc), but it's because, for most of history, people are surrounded by people! For a long time! And da's a culture, babeeyyyyyyyyyyyy~
As cultures got larger and mingled with others, became melting pots, and our ability to engage with other cultures grew via telecoms, it became clearer that culture isn't strictly geographical, but only partly.
When we can go to groups outside of our community, or find audio-bearing media that, more and more, exacerbates the accent as a trope, or go online and find distant culture, we might find parts of ourselves we desperately want to know, which can add a bit of oomph to the adoption of a mannerism - you like a thing, a person or people even, you act more like them - even unconsciously. It's noticeable even just in conversation - people can sometimes unknowingly match the stance and energy of a person they're talking to, usually if they have a really high opinion of that person, again, whether they know it or not.
So, it'd be my guess that it IS just cultural, at least, in terms of adopting the accent - it's just coming from Gay Culture. The thing I don't know is where it came from within gay culture, but I wouldn't be surprised if Britain had a hand in it.