I am by no means an expert on this, but we did just have to move our 6 yr old mid-school-year. This is what I experienced.
Some schools handle their own admissions. Some school admissions are handled by the council (county). We were not able to apply to move our daughter until we had atleast a job contract and a rental contract showing which catchment area we would be moving to. We had to submit these with our application to the council, along with a list of preferred schools ranked 1-3.
I did contact the school we wanted directly and spoke to the head teacher, and confirmed that they had room for her and that our new address was, in fact, in the catchment area. The headteacher told me there was not any reason she could see for us to be denied a spot, so that was helpful. The council we are in did have a ten working day reply time.
Ofsted is one site you can check, if you haven't already, to see what the school's last inspection results were and how they are currently graded.
Hope that helps, good luck!