My children were born in the early 90s so I remember this as a parent. Every part of England did it differently. In our Inner London borough, you could apply for as many primary schools as you liked. We applied to eight for our daughter as we knew we were in a black spot and we only got one offer.
Each school did its own admissions, applying a published admissions policy, usually giving top priority to siblings and then going by home to school distance, sometimes with priority to children in certain areas. Obviously different for religious schools.
Children started Reception the term in which they would turn 5. My daughter was April-born but the cut off was 30th April so she started in January. By the time my son started Primary a couple of years later that had changed so children born between 1st September and 28th February started in September and the rest in January.
Reception and Key Stage 1 class sizes were capped at 30 by the Labour government in the late 90s. In our borough it was unusual for KS2 classes to exceed 30.
By the time my daughter was secondary age, we had co-ordinated admissions for the whole of London. You could apply for up to 6 schools. You were offered a place at one school or no school. All applications for each school were processed independently and ranked against the admissions policy criteria. Then all the lists were amalgamated and a hugely complicated computerised process begun of re-ordering the lists based on parents' preferred rankings.
Say Sam's parents had applied for places at schools A, B, C, D, E and F, ranked in that order, and Sam was ranked 321 for A, 345 for 2, 667 for C, 122 for D, 29 for E and 112 for F - and say they all took 240 children a year (unlikely) - Sam would be in line for a place at D on the first computer run. But the computer would then remove from A, B and C's lists all the children who came high enough on the lists for other schools that their parents had ranked higher that they didn't need their places at A, B and C. All lists were then re-ordered, again and again, until all places at all schools had been allocated. That might well mean Sam ending up with a place at A, B or C in the end.
This was a far better system than what had happened previously as some parents had multiple offers from different boroughs, grammar schools and academies/city technology colleges, and other parents had none at all.
Also, the Admissions Code of Practice had come in and schools were no longer allowed to interview parents or children as part of the process. Parents and LEAs could appeal against admissions policies they thought were unfair.