How state school admissions work (for Grey Coat, St Marylebone and all other state schools in England):
Schools have a list of admissions criteria which they must stick to rigidly, which may include children in local authority care, siblings, distance, selection exams, faith, etc.
Parents list the schools in their genuine order of preference, but it is always a good idea to list at least one school somewhere on your form that you have a very good chance of getting into, usually your nearest non-selective school, even if you're not too keen on it (as it is better to have a school you don't like nearby than one you don't like that you need to travel to).
Through this, you have put in an application for every school on your list, whether it is first, sixth, or anywhere in between. At this point, the position does not matter at all.
The school or the LEA list all applicants for a school in order of how well they meet the admissions criteria, with no reference at all to what preference they are. If (hypothetically) a school's criteria was distance only, and you live nearer than me but have put the school sixth and I have put it first, you would be on the list ahead of me, as preferences are not taken into account at all as long as the school has been listed somewhere.
At this point, for each school a line is drawn on the list at the number of the PAN (published admission number) and everyone ahead of this line can be offered a place.
The LEA then looks at the schools that can offer everyone a place. Some children may appear above the cut-off line on more than one school's list, so it is only at this point that the preference order comes into account. Everyone can only have one offer, so you will be offered a place at the school you listed highest (so if the schools you listed third and sixth can offer you a place, you will be offered the third choice and crossed off the list of the sixth choice).
Due to some children being in multiple lists and only being able to be offered the school they listed highest, they can be crossed off the list of any schools they could be offered but are a lower preference than the school they have been offered. They are therefore crossed off these lists and everyone else moves up a place.
Eventually, this process is complete and everyone has been offered a place at one school- their highest preference for which they could be offered a place.
What definitely does not happen (though is a common misconception, and the way it worked many years ago) is for a school to receive a list of everyone who listed it first and offer places to these children before moving on to look at those who listed it second, etc.
I may have over simplified the process somewhat, and I'm sure the admission experts can correct me if so, but hopefully this gives you the gist.
Apologies for the essay and the multiple posts in a row!