When it's a question of the rights of the child versus the wants of the adults, in UK law the rights of the child will win. The can of worms should the legal definition of 'mother' change would be huge, and would open up all manner of ethical implications.
That might be coming, though. The lawyers arguing against this are couching it in the terms of the rights of trans gender people rather than the rights of children to know their own parentage. They claim trans rights are 'piecemeal' and only applicable in some circumstances, which is only a logical conclusion if 'gender identity' is conflated with biological sex.
Asserting these rights at the expense of women is one thing, children quite another. UK courts still always prioritise the latter. Unfortunately if this movement continues gaining traction I can see that changing.
Hope this isn't just a temporary victory.