That's great that your private school is one of those who have agreed to pay.
But leading on from your initial comment where you were unhappy about the requirement to pay, you do understand now that state schools pay for it too?
In your previous post you said it was unfair that the private school has to pay - but state schools also pay. No one gets it for free. The payment system is just less visible to parents in a state school.
I was a school governor for 5+ years, hence having visibility of finances, arrangements etc that weren't available publicly.
I have two SEN DC and unfortunately in our area, private schools are spectacularly uninterested in providing support - unless you also happen to be academically gifted, because of course, that would look good on their stats....