There is so much advice on here that is legally correct but absolutely useless, because the system is totally broken.
Yes the LA have to legally place your child in an environment that can meet his needs.....but there aren't enough places. And appealing, going to tribunal etc doesn't change that.
One thing is certain though - you don't want a place that says they can't meet need forced to take him....that is a recipe for disaster.
I would do the following-
Check the ehcp is correct and up to date, and make sure that the schools are making their decision on the proper information.
Check with nursery how long he can stay with them, and factor that in.
If the private school is a good option then enlist legal help to force the LA to pay the fees.
Join every national and local group on Facebook you can regarding ehcps and find real time support there.
Write to your mp today highlighting all of these issues. Copy this letter to the head of every school he has been refused from, and the opposing political party candidates in your region, along with the head of local SEN services, and the press if you are so inclined. Make a fuss. This will not get sorted anytime soon unless more places are created.
Have you had an assessment by children's services disability team yet? You are legally entitled to one for your son, and a separate one for you as his carer. You may be able to access some support and respite there.
Stop worrying about where your daughter will go to school, that's a few years away, focus on this first.
Consider the possibility you may need to move to be closer to a school that will meet his needs and take him.
I know all of this is heartbreaking stuff, but it's the reality at the moment, and as Sen parents we have to we have to focus on meeting our kids needs, whilst the state won't. It's utterly shocking and I wish you luck xx