carbfuelled,
I'm sure you've already researched this thoroughly, but it is probably worth saying in case you haven't: if you are an 'in year' application (ie not one for Reception at the normal time for all admissions) you don't really get to choose schools UNLESS you find that several have a space for you.
If a school DOES have a space and no waiting list for that year, then you can be assigned that space wherever you live in the country at that point IF you can start almost immediately (we did this for DS - we were relocating, but DS's was a low birth year so many primaries where we were moving to had spaces. We visited several, and were given a space at our preferred school even though at that point we lived in our old house. We had to start within 2 weeks, though - in the end I took him by car while our removal van followed a bit later in the day!)
If, as is more likely, the best - or all - schools are full, then you have to apply to them all (councils have slightly different rules), and will be assigned a position on the waiting list in line with their general over-subscription arrangements (so, for example, if you move next door to a school with 'distance from school' as the main tie-breaker, then you will jump to the top of the waiting list). Then you wait... meanwhile...
It may be that one or more less popular / highly rated / more distant schools have spaces. Unless you wish to home ed or send to a private school, then you will need to take one of these as an interim measure.
You may very probably have to appeal for several schools if all schools in you locality are full OR you are temporarily given a space in a school you don't want. You will be given details of how to do this if all the schools you ask for a place at are full.
The Fair Access Protocol can be invoked if you are at risk of having no school place at all - but this forces you to be given a place at the school best able to cope with another pupil, NOT the school of your choice. This is a fairly common misconception that you can ask for a place at a school you want under the FAP and expect to be given it.
How old is your child? If Y3 or above, your life is somewhat easier as appeals can be won on balance of prejudice - whereas Y2 or below the Infant Class Size regulations apply, restricting schools to 30 per class unless there are very specific and quite extreme circumstances.
So don't pin your hopes on one school (though you may be lucky - places like Cambridge will by their nature have somewhat transient populations so in-year places may come up more regularly than elsewhere) - rather find out a wide range of schools that would be 'OK' and research which of these, if any, tend to have in-year spaces. The LA should give you details of any school with a place at this moment, though this will change daily, and may be able to advise on number of in-year admissions per school per year.