It's already spread everywhere.
I think the culture of where it spreads to does change things, though. The UK is derisively called TERF island, and it's not necessarily because it's full of frothing at the mouth transphobes. The public figures challenging the extreme rhetoric are lefty liberal educated middle class types with townhouses in Islington, whereas their counterparts in the Upper East Side seem to stay quiet.
There's definitely a strong contingent of people who are sold wholesale on it (after all, Aimee Challenor got high up in two political parties and the largest LGBT campaign group in the UK despite being only 22/23). But I never
quite saw the same level of hysteria as I saw coming out of the states. People would have similar viewpoints and be aggressive about it, but they wouldn't
lose their shit in person in quite the same way. It might have changed now, it's been a while since I was at university. There's maybe more of an aversion to people screaming and taking themselves too seriously - on the opposite side, they tried airing Fox News over here and it never took off, because even though Glenn Beck's views may have chimed with some people, they looked at him yelling and crying and went "who's this prick?".
I posted this
YouGov poll here the other day. It was initially hailed as a triumph - a plurality of Brits in general support the statements of "trans women are women" and "trans men are men", and support the right to self id. But there's a hardline of thinking trans women shouldn't compete in sport, or use women's changing rooms pre-op. Some of those views hold across the most woke demographics. I'd love to see comparative polls run in the most liberal parts of the US vs the US overall. It's hard for me to gauge exactly what it's like in some American cities.