Does changing schools affect the Non-O or ED visa status in Thailand?

You cannot stay on the current extension based on the student/guardian when the student changes schools. Those visas/extensions are tied to the school.

When the student stops studying at school A, you cancel extensions of your child and you, with the termination letter from school A. Then apply for Non-ED/Non-O (visas and/or extensions) with school B, where the student continues to study.

It depends on both schools' procedures whether you can apply for new visas and/or extensions with school B immediately after cancelling extensions with school A, OR you must go out of Thailand once to apply for a new Non-ED/O visa. If both schools collaborate well, you might complete the changing procedures in Thailand without leaving.

You need to talk about the details with both schools well in advance.