Birth certificate (illegal??)

[CheeryDeer]

UK Based
Hello, hoping for some advise for a friend.
She had a baby and recently got the baby registered, she's currently going through postnatal depression. Heparterner is trans (female-male) and during the registration she left him to do most of the talking, anyways, he put himself down as the dad on the birth certificate. They used an anonymous donor to have a baby together.
She's worried if this will ever get found out and what would happen if it does.
Any help would be great, thank you!

Advice:
The risk is that the trans parent may have no real legal link to their child. The implications of this could be huge eg they could have no say over life and death medical decisions, the mother could take the child away forever, etc etc. That isn’t a risk I’d want to take with my child.

Better to get it sorted legally - The result of which might mean the trans parent is on the birth cert as the other parent and nothing changes, or it might mean the need to adopt, or something else entirely. First step is saying they made a mistake.

What do you think they should do @CheeryDeer?

Edited to add: x-post (I lost Wi-Fi.)

Interesting that it can be amended (which is a good thing) but not that of the child who was accidentally registered as the wrong sex. Or was that one sorted out in the end?
Either way, if this kid ends up with a truthful document, all's well that ends well.