How the admissions process usually works:
The schools have a list of criteria, in priority order.
They rank all applicants according to those criteria. They have an ordering sub criteria which is often straight line distance to the school.
Where a child can be offered a place a multiple schools the LA accepts the one with the highest preference and then removes them from the list at lower preference schools, then checks the first child on the waitlist at the lower schools to see if they had 2 offers now. Once all movement stops you have the lists for offers day.
Everyone receives their offers, you have to accept or decline fairly quickly. Most schools also publish a list at this point of how many places in each criteria were offered and what the distance was if they were oversubscribed.
SUPER IMPORTANT - accept your place unless you would rather homeschool than go to that school, it has no impact on the wait list
A couple of weeks after the deadline, second offers day. This is when late applicants get added to the mix, basically everyone who rejects their offer on offers day has their place offered to the next person on the list.
Then waitlists start on a one out one in basis. There will be movement all through the summer, there will be movement all through the first year. If you get past the transition day have a long think about if you got offered a move would you take it, when your number comes up you will not have much time.
It's incredibly difficult to win an appeal, unless there has been an error. Social and medical grounds sounds nice, but realistically you need a doctors or social workers report naming the school and stating the harm to even stand a chance.
And don't give up. I got offered a place at my preferred school last week, and there were 3 other new students coming in at the start of this half term. Sure he's missed a term and a half of learning, but he's still got 6 and a half years to go!