Hi let me state by saying I'm afraid I do not know the school admissions appeal procedure very well. It is also worth saying that different regions use different criteria for allocating school places. I do however know that it is possible in many cases to "politely" bully these local authority groups. From what you have said your child is going into school at reception year. This would mean that they are starting with all new children (important as they cannot say the children who have been put above your child are already within the school system.) It seems that you have quite a few things in your favour such as the fact you already have children in the school, the fact I assume you live near to the school and the fact that you have no transport.
Now most schools in my area have n intake of around 30. If your area is the same (which I am assuming that broadly speaking it is) then what they are saying is that 30 children have more of a need to go to that school then your child. If I was you I would contact the admissions line and ask them for a detailed reason in writing of why your child has been denied. Then tell them that if they are seriously telling you that your child is not likely to get a place then you need them to fully document WHO was responsible for this decision as you will be taking your case to the Local Government Ombudsman. This is the body to make sure that they have followed their own guidelines which my guess is they haven't.
I would remain calm throughout and keep stressing (the truth) that all you are is a concerned parent who wants the best for her child. Also stress the hardship it will place you under and the fact that you will have to rely on the free transport to and from school for ALL your children due to this decision. I would also stress that none of your children are able to go to school with anyone else, so in effect they are asking you to choose who's education is most important to you.
At this stage I would also ask them what they suggest is your best way of getting your children to school in the event of the appeal going against you. Ask for this in writing and ask for it to be signed by someone in there department.
What all this does is several things.
1 it shows them the problem in real terms, real people, not just as a name on a screen.
2 It shows them the financial cost to them of your child going to a different school.
3 It shows them that you will hold an individual responsible for the decision (the individual who is now deciding where your child goes) and if it is found that that individual has given a place to someone less deserving of a place then they as well as the group as a whole will be accountable. There is in my experience nothing local authority people hate quite like being singled out for the possible sack.
When you do all this it is (in my opinion) very important to remain as calm and reasonable as you can so as to give them a way out (giving your child that school place you want.) Just firmly show them that you will continue to fight for your child's education until they listen.
I can't guarantee this will work but in my experience local authority employee's are very paranoid (with good reason) about losing their job for not following the rules. By doing this you are making it personal to them so in their minds it's not just your life that could be effected by their decision. I have used a similar process on adult social services to get a friend of mine help with her dementia suffering husband.
Like I say I am not telling you this as an expert, just as a fellow parent. Hopefully other people have more experience of this procedure then me and can give you more advice.
I hope it goes well. let us know what happens
Just to be vey clear and transparent, this information is widely available to those who are not in the legal profession and I am not telling you anything you could not (and most likely have) found out for yourself.