I tried to compose a post to lay out how it might be workable but have given up because it is not.
Prerequisite for even trying would be a no deal brexit as any deal that had us still following EU rules would prevent this anyhow.
Nb1 - it would be completely impractical for most private schools to become state schools without a lot of building work as all the classrooms are built to take 15-20 students rather than a state school's 30-35. Also computer rooms, science labs etc all only kitted out to supply half the equipment needed for running as a state school.
Nb 2 - You can't do this without throwing the kids of military and other international workers under the bus, along with a lot of SEN kids. The former, plus the very wealthy, will decamp to a nearby overseas location instead so it's SEN kids that will be worst hit.
But you can get around the assets issue by getting each private school to create a separate charitable academy trust to proceed as a state school. They would each need an admissions policy based on random lottery with no catchment area, and would obviously need to sack half their staff and sell off some of their buildings and land for development of exclusive 3 & 4 bedroom homes in order to function on the £6k per child from the government rather than £15k from parents.
Obviously yes private medical treatment needs the same kind of policy. Also private nursing homes for the elderly. Private sporting facilities are way nicer than council ones too. There are vast swathes of industry that need destroying for the common good.