Headmistress didn't back down
My daughter attends a local private pre-school. The parents of a boy informed the school that he now identifies as a girl, and that the teachers and other children should now refer to him with female pronouns and his new name.
Fortunately, the Director of the pre-school is a very experienced woman who owns the school herself and has a background in child protection. She responded by saying that every child's voice is important and she would not be compelling the other children to use that language or to confuse them (these are 3 and 4 year olds!).
Also that the toilets are open plan and mixed sex, seeing a "girl" with a penis could be distressing and confusing to the other children. And we all know who the boy is, it’s the big boofy kid with the male nickname in a Frozen dress.
The Director also told me that she was contacted by a private organisation that said she needed to sign up with them to provide training for the teachers regarding gender identity, gender non-conformity and LGBTIQ sexualities under the guise of “diversity, inclusion, equality” training. When the Director refused, the organisation (wrongly said) that they would be legally liable if they did not. The Director is an experienced and mature woman who would not be able to be pushed around, and she stood up to them.
As far as I am aware, they did not contact her again. But imagine if it had been an inexperienced young manager who did not feel she could stand up to this person?