This happened to me a few years ago. I went to appeal and lost and had to wait a term until a place was available. Sadly, with three children it's unlikely that they'll all get places any time soon.
You might be better posting on Mumsnet, where a couple of appeals 'experts' post. Unfortunately, I expect you'll get the same answer as I've just given you, especially if any of the children are in KS1.
On a positive note, if the children are still at their previous school and it's six miles away, the local authority should pay for transport (taxi?), unless there's a nearer school with places.