Unfortunately, what usually happens is that they just fail to turn up on the first day of term. You then spend the next fortnight trying to get a response from them (fortunately, most do actually get back to you and say 'Oh, they're going to x Private School'), then waiting for the first day of term at that private school, getting confirmation from that school that they've turned up, then offrolling them on that day - they stay on the register by law until that day - and then you chuck away the ID passes, get finance to close the payment account, get IT to remove the Google Classroom and network access, get each department to remove access to online resources that they manage, raise a digital transfer file and upload it to the government data transfer site with the appropriate code, complete a migration report and submit that to the LA, refresh your waiting list, offer the place(s) to whoever is top, wait for them to respond, chase them if they don't respond, chase them again, ask the LA to separately contact them and chase when they don't respond, then remove them if they do respond and say no thank you, sort out a meeting for them to come in and decide whether they want the place after all, manually add them to the register, arrange for the payment account, online accounts, added to classes and timetables, send details of uniform, timings, etc, notify reception to expect them, package up the hard files sent by the primary and post them to the private school, chasing for acknowledgement, arrange for the DSL to transfer any Safeguarding hard files, arrange for the SENDCO to transfer any SEND information....
...and then you repeat all this for the other nine kids that haven't turned up.
Except where you can't get a response from the parents or they say they've decided to home educate. You then have to complete a five page referral form for the local authority detailing every attempt to contact, submit that after two weeks of trying, communicating with the previous school if they've included details of them on the original application (not all do), then they do as much as they can to trace them and eventually respond sometime between mid October and Christmas (as they have hundreds of CME cases, along with children genuinely at risk of something awful having happened in all year groups) and tell you if it's a case where you can remove them from the register or if 'it's an attendance issue', which means it's your problem.
-------
Seriously, private sector parents, just decline the state offer now. The amount of work it causes, the cost in time and physical resources and, most importantly, the sheer noise it creates directly affecting the urgent safeguarding that has to go on for children at genuine risk is ridiculous.