Sounds different from school to school as my experience differs from several posters here.
We have a 60-odd page gender policy document, which has been approved by governors, including parent-governors. Staff are not required to read it in its entirety, but training on it is/was compulsory.
Again, unlike other posters, where students ask for us to use a different name/pronoun, we are explicitly told not to refer to them as such until meetings with the family have taken place. A student can't just declare their name changed. Of course, we are sensitive with said student, but it's remarkably easy to avoid pronoun use and do the register by eye (secondary school), until such an issue is referred and dealt with. It is always a senior teacher, usually a deputy head who liaises with the family and will ultimately tell staff of name/pronoun change, if it comes to that. None of our trans students have ever reverted back to their 'previous' name/pronoun. Still have only encountered this with a tiny handful of students in the last 10 years.
We have a third space for changing/ toilet.
To answer the OP's question, I don't know about staff support if they don't agree because I don't know what a teacher could reasonably disagree with about the above approach.