For Women Scotland wasn't the start of the opposition to gender identity policy in the UK. It was founded, by four women who met on Mumsnet, specifically to address policy in Scotland.

Feminist women opposed to the Tory government's plans to introduce "gender self-id" law and similar policy had already started organising by this point. Groups like Woman's Place UK and Fair Play For Women. This had nothing whatsoever to do with religious arguments from the US.

There's also significant liberal opposition to all this in the US, again not linked to religion but, like the UK, on the basis of women's rights.