You can appeal, but if there are no places available, its unlikely you will win. If its what they call a class size appeal, its just a formality. You must prove that the admission criteria was not applied properly in your case, its very hard to win an appeal. I only know of one person who has ever won an appeal. That being said, I am sure some people do win appeals.
We appealed and lost and sent our son to a different local school, and it was the best thing we ever did. He loves it and we are very happy. Its a little bit further than the one he was in the nursery at, but it doesn't matter as much as I thought it would. My suggestion is look at some other schools, and see if they have places. Good luck!