Sorry to hear that your appeal was turned down.
I act as Clerk to the Panel in my Borough and we have a lot of parents that appeal because their child did not get a place as the parent did not apply, or made a late application. The Panel are normally not sympathetic to such cases, especially if the parent has an older child at school, for whom they would have had to go through the admissions procedure already.
In the Authority I live and work in posters to remind parents to apply for a school place are visible at GP surgeries, play schemes, nursery, bus stops etc. The Authority does not automatically send out a pack as it is seen as the onus on the parent to find out and apply for a school place and the Local Authority just has the obligation of offer or notify of a school place once the application has been received.
AS your child is of reception age, Infant Class Size Legislation applies i.e. max of 30 children per class. Your case has to be extremely exceptional for the appeal to be allowed or you have to prove that the Admissions Authority did not apply their admissions criteria correctly. Even if you have an older child in the school, although sibling priority may apply there may be other siblings also waiting for a place so your child will not necessarily be first on the waiting list. Also waiting lists change daily i.e. in my Borough we don't have primary catchments, so its based on distance, so tomorrow someone else could move closer, one sibling gains a place in say Year 3 and his/her sibling ends up on the waiting list, pushing your child down iyswim.
I am afraid that unless you can prove some kind of maladministration there is not much you can do. If however you do feel that there has been maladministration then you should write to the ombudsman giving details of your case and they will investigate your individual case. Contact the Clerk to the Panel and request a further action letter.
Sorry to have gone on and good luck.