You can go in waiting lists for other schools straight away.
Many areas will automatically put you on lists for schools higher up the form for which you did not qualify in the first round. But you need to check, and you can ask to be out on more waiting lists as well.
It is your right to appeal.
But appeals where infant class size (ICS) rules apply can only be won in three ways:
a) admissions error which meant your child did not get an offer they would have received if they had administered the system properly (so for example, have they overlooked sibling? Or measured distance to your house wrongly?
b) the criteria are unlawful, by not giving priority to a group they must, or giving it to a group they must not. (This does not mean schools have to use all possibles of the optional permmissible categories).
c) the decision is so unreasonable it is legally perverse (barrier for this very high, eg child protection issues).
That they offered you the nearest school with a space is entirely correct. It's not a synonym for nearest or next nearest. Is it one you viewed, or do you know anyone with a DC there? Finding out more about it might be a useful thing to do.