Happy I am sure there will be a push for that. I think the thing that holds it back is that a mother must be the person who gestated the child. So for instance a lesbian couple cannot both be registered as mothers. One can be the second parent but there can only be one mother. Under the current law, a biological male wouldn’t be able to claim he was the mother. But i do wonder for instance if surrogacy law is changed so that the woman who gives birth is no longer the mother (as is the case now) whether a transwoman will try to argue that they should be listed as the mother.
Also, if the desire to change the law is pushed through as I am pretty sure it will (as some of them have promised their funders that is exactly what they will do) then it becomes a free for all. I think one of the projects, the legal gender one, is more about neutrality, so no reference to sex on birth cert for baby or parents. OTOH, Sally Hines seems more in favour of affirming people’s ‘inner identity’, meaning we could have male mothers and female fathers. Both examples would be bad but I would prefer the neutral one as at least it doesn’t distort reality to the same extent as the gender identity approach.