It must depend on area. We have a local Children’s Centre that specialises in speech and language and OT and some of it is delivered in the Centre, some in state schools and I think for private school kids they go to the Centre.
Just to balance this out a bit, private boarding schools, the most expensive ones as well, do get NHS treatment via GPs I think and allocated hours as such so the kids there quite often get to see the GP quite easily. Also in our area nurses definitely visit state and private schools for vaccinations.