I thought the law stated that the EHCP must follow the child, so then at least for the first year, they need to base provision on the existing EHCP
No, not quite.
When you move LA with an EHCP, the new LA becomes responsible for the EHCP on transfer. The EHCP transfers on the day you move or within 15 working days of the LA becoming aware if that is later. This includes funding the placement in section I if attending is still practicable. If it isn’t practicable for DC to continue to attend the named placement, the LA must make other arrangements for education, including the provision in F. That is in Reg 15.
However, within 6 weeks of the date of the transfer, the LA must inform you when they will review the EHCP. This must be within 12 months of the previous AR or 3 months from the date of the transfer, whichever is later. So, no, you aren’t guaranteed the existing EHCP for at least a year. As I said, the LA may propose to amend or even propose to cease to maintain. You would have the right of appeal, but it is something you need to be aware of. Particularly with the waits for appeals. They may also undertake a reassessment of needs.
I spoke to the headteacher of the independent school who said they aren't permitted to say whether they would accept the child until the LA asks them to consult
Some independent schools won’t offer a place until formally consulted. But they can if they want to, so it isn’t quite true for the HT to say they aren’t permitted to.