I have a fairly specific question about primary school admissions in England. Suppose that we move to a new area with child that is already school age (e.g. move from one part to another or move into England from abroad). What are the rules/chances to get him/her to a state school in that area? Assume that there are many primary schools nearby (e.g. in London).
In our area of London (Southwest) it depends a bit on the age of the child. The entering classes are over-subscribed (ages 4-6), so it's difficult to get a place, but the local authority will offer you something. It may not be in your first choice school and you may end up with a commute. Above the age of 8 or 9, families tend to move to where the secondary schools are good, which means that there are often extra places available in the good local primary schools. Our local population is rather international and mobile, so there are a couple of leavers and new children each year.
The new allocations start in January for the following September. The entering class places are offered around April, but there is often a lot of change before school starts in September. You might have better luck getting a place if you can apply in the spring.
I believe that your local authority is under an obligation to provide a school place -- and that's all.
Everything else depends on the local authority. For instance, our local village schools are way oversubscribed so even if someone moved next door they would need to travel four or five miles to the only school that has places available.
Some local authorities will provide free transport if the school place is beyond a certain distance from home - typically three miles.