Can a mother change the child's school (without consent of the father)?

[UnsureOfWhatToSay]

I am posting on behalf of my partner. I hope that's okay.

My partner has an 8-year old child. The child recently started year 4 at school. Both parents currently live within walking distance to the school. My partner specifically rented a house near the school to make drop-offs/ pick ups easier (this was before he met me).

My partner's ex told my partner tonight that she is moving to another area. This area is about a 20-minute drive from where she lives currently. My partner and his ex are not on friendly terms. This communication came through Whatsapp. My partner didn't respond as the message was sent late-ish (10pm) and my partner believes that his ex may be drunk and trying to get a reaction from him. He will ask for more details tomorrow.

My partner is now worried that his ex will try to move the child's school, which my partner does not want. Is there anything my partner can do if she does try to do this?

We are in England (if it matters). There is no court order in place.

I got in writing only via SMS
then he asked me to sign agreement by certain time and few hours before the deadline, he called the school and admissions.
in signed agreement he is forcing me to cancel CMS and reduce his payments by 50% which I have to agree to (he said he will put in writing he will be paying as once he stopped in past)
so agreement is signed but I still feel he is blocking me as he still asking for some other things, so never ending….
I have a log of his time with son and phonecalls
In agreement he is asking me to remind child to contact him while it should be his responsibility, 90% of the time is my son asking for phone contact
I don’t want to spend loads of money going to court unless is necessary