Hi ladies,
Was just browsing the comments as I'm in a similar boat, since moving from one borough to here, my children are still awaiting a space in this borough therefore we have an annoying commute everyday.
Only thing on your part, I would also suggest writing to the admissions team expressing the urgency of the matter and even copying in the local mp if needs be as your child has no education at all.
If you let them know you can afford private schooling I doubt they will see your case so urgent.
Whatever next step you choose, I wish you every success!
Take care