Swissrole - Starting with your older child, as there are already 30 in the class and it is year 2 any appeal would be heard under infant class size rules. That means you can only win your appeal if you can show that the LA has made a mistake. In your situation that is unlikely. You would have a better chance in year 3 when class size rules no longer apply.
For your younger child, however, you have a better chance of success. As there are only 24 in the class, infant class size rules do not apply. That means you only have to show that the prejudice to your child through not being admitted outweighs the prejudice to the school through having an extra child. As they are already operating with 30 in at least one infant class and the sky hasn't fallen in, you can use that to suggest that one or two more in reception won't be a huge problem. That won't guarantee success but, particularly if you can make a good case as to why this is the right school for your younger child, you will have a reasonable chance.
Luciemule - Schools do NOT keep a few extra places for travelling children, forces children, etc. That would be against the law. Every Local Authority has to have a "Fair Access Protocol" which says how places will be found for certain categories of children including traveller children and armed forces children. It also includes children moving into the area outside the normal admissions round whose parents have been unable to find a place for them due to a shortage of places. They will normally be placed at the nearest school with places available. However, if there is no school within a reasonable distance with places available they will normally be allocated to the nearest school best able to accept additional children, even if that takes them over the class size limit. For one year these children are "excepted children", which means they don't count towards the class size limit and therefore no laws are broken. After that the school will have to employ additional teachers to get the classes back down below 30.
Your school is also wrong about asking a parent to remove their child from the school to give a place to another child who lives closer. They are absolutely not allowed to do that. The Admissions Code is clear. Once a child has started at the school you can only take the place away if their application was fraudulent. If they try to take a place away from a child simply to make way for someone else they will be in all sorts of trouble.