OP ? from what you have said:
?One of the children just wanted to move from a school nearby and went to appeal, they won, despite the classes being already huge at our school.?
If someone has won at appeal there is nothing the school or LEA can do. However, it would be very unlikely that anyone else could win on appeal if you now have 32 in an infant class.
?Another child lives miles away and I know for a fact that there are other schools which are not oversubscribed and are closer - I know ours is a good school but surely that cannot be reason to appeal.?
This sounds like a child with SEN or a looked after child (ie a child in foster care or adopted), given you state that they do not live nearby.
As tiggytape says, the school, governors and LEA are in a difficult position if you write to complain re this admission. They cannot divulge confidential info to you re why this child was admitted, they can only say that they were required to admit the child under LEA admissions rules or similar.
I would expect the school/teachers will have the same concerns you have and will be looking at what can be done to alleviate the problem.
I expect PRH will know this but I think in this sort of situation the school is required to come up with a plan for the next school year re how the class can be brought back to within 30 children. This could be anything from hope someone will move, do some mixing up of age groups, or put another teacher in. Unless this is a Y2 class that will then move to being Y3 which is not covered by infant class size restriction.