That is clearer. Thank you.
I was under the impression you had given all the evidence about your daughter in May and it took them months to arrive at a decision during which time another child was admitted. Indeed, I was under the impression that Surrey have already agreed to put your daughter in the "exceptional arrangements" category.
Looking at your latest posts it seems that Surrey have still not agreed to put your daughter in the "exceptional arrangements" category and that you did not provide the letter from the specialist giving the mileage until after this other child was admitted. If that is the case they have behaved correctly. At the time the vacancy arose this other child was ahead of your daughter on the waiting list. The fact that you subsequently provided the letter from the specialist which they requested and which may mean that your daughter will be moved into the "exceptional arrangements" category does not change that.
What matters when a vacancy arises is who is top of the waiting list at that time. Subsequent changes to the waiting list are irrelevant. If Surrey had taken an unreasonable time to process the evidence you provided there may have been an argument that your child should have been at the head of the waiting list. However, on the dates you have given they didn't even have the required evidence from you when this vacancy arose. They therefore had no choice but to offer it as they did.
I would like to help and I've been trying to find evidence that Surrey have broken the rules in some way. I am afraid everything you have posted to date indicates that Surrey have got it right. Y4 is full and therefore you have to wait until someone leaves. If they do put your daughter in the "exceptional arrangements" category she is likely to be at the head of the waiting list and will therefore be first in line for any place that becomes available in Y4 but they do not have to admit her until a place is available.
Having said that, if they do put your daughter in the "exceptional arrangements" category you could suggest to Surrey that circumstances have changed and you should be entitled to a fresh appeal. They may not agree to this but if they do you can use the letters from the specialist as part of your case. You may be able to persuade an appeal panel to admit your daughter on medical grounds. But if they do give you a fresh appeal please concentrate on the medical evidence. Everything else is irrelevant. If you go into an appeal with the scattergun approach you have used here the panel is likely to end up confused.