Unfortunately, it seems that you have been poorly advised by the nursery, as the best way to guarantee a place in a particular school for a child with SEN is to have it named on an EHCP.
There are a number of things that you now need to do in parallel:
Check that no error has been made. It being your nearest school is only very rarely a factor, and that only where it is explicitly stated in the over subscription criteria. Much more likely is that there were 30 children who met the criteria better than your child (so had EHCPs that named the school; were looked after children, were siblings, lived closer) - but do check the last admitted distance is actually shorter than your home-schooll distance.
Go on the waiting list.
Get the EHCP process moving fast. All the extra information you submitted with your application actually needed to be in an EHCP application instead. While you are not 100% guaranteed a place if you get the right EHCP (for example if there are an overwhelming number of children with EHCPs already admitted, it is just possible that the school may say they cannot meet need) it is extremely likely. Getting an EHCP is time consuming, onerous and can be adversarial- expect to appeal and if necessary refer for multiple reviews.
Contact the allocated school and have detailed conversations and share all information with the SENCo there, including potentially them observing your child in their current nursery. The better prepared they are, the easier your child’s transition will be if this ends up being their school.
Prepare for an appeal. As an infant class size case, you have to prove that it is something legally unreasonable (a VERY high bar) that deprived your child of a place - not the much low bar relevant for Year 3 and above where it is balance of prejudice to the school compared with your child. Go into it in the knowledge that you are unlikely to be successful, but it does happen.