The oral pronouncement of the judge is a valid court order. Purposely failing to abide by a court order "could" result in contempt of court. Enforcing a court order that didn't make it into writing may be difficult for obvious reasons. You should seek a modification to be safe.
Do not disobey the court's order, the consequence could be severe, the better thing is to go in for a modification and explain your situation to the judge. You may also want to get a supportive letter from the school staff. If you do it on your own you risk contempt charges, including possible loss of custody and a fine?
Legal disclaimer:This message does not constitute legal advice and does not create an attorney-client relationship. Any statements are made for general informational purposes and do not constitute legal advice. No attorney-client privilege is created by this communication. Attorney is licensed in California only.
See a local lawyer. you may be able to seek an ex parte temporary order pending modification hearing, so the kids can go to the proper school.