You want someone to explain the school system in the UK?
I can only speak about the primary school system. Things have changed recently.
You now have three kinds of state school. Local authority/community schools (these can be religious schools), academy (run by a private sector organisiation), and free schools (really new, not sure what the definition is yet - first ones opened in this school year). The latter two are not answerable to the local authority but directly to the Dpt of Education.
Then you have independant schools (fee paying), and public schools (fee paying).
To apply to a primary school, you need to fill in a form with the local authority listing in order of preference the three schools you want to attend.
Most primary schools that are under local authority operate on proximity to the school as their only entrance requirement. However there are others that get priority.
First priority goes to children in care.
Second priority goes to children with siblings already at the school.
Then children get into the school according to how close they are until all the places are filled.
The school does not know what your preference is. All applications are treated as the same. If you get accepted into two or all three of the schools on your list, then the one that is your top preference is offered and you're taken off the lists for the other two without even knowing that you were accepted.
Where a school is oversubscribed, possibly because there aren't enough schools in the area or it is outstanding, or it is a very small school, then there's a chance you won't get in.
If you aren't accepted into any of the three schools on your list, your local authority is obliged to offer you a place. This will be at the nearest school with a vacancy - which may not be at all that near you, and is likely to be a poorly run school with low results.
That's the theory. In practice I know of at least two parents who did not get any of the their preferences, and were not offered an alternative by the local authority.