So what happens if more children score 121 than there are grammar school places available?
Essentially this doesn't arise, because there are ample GS places for qualified children in general terms. However, when there are more qualified children than places for a particular school, the over-subscription criteria are used to determine who has priority for places. Those criteria are (very roughly):
1. Children in care and those with statements naming the school.
2. Siblings (rules vary a little on the definition of that)
3. Children living within catchment for the school.
4. Distance from school gates.
The criteria are applied in descending order. There are very few schools that are oversubscribed for children living within Bucks - Challoners Boys, John Hampden and Royal Grammar are the pressure points, i.e. boys in South Bucks. Most other grammars are usually able to accommodate virtually all of their first preferences from in-county.
Does a child who scores say 70/80 on both papers (for example) have more chance of achieving >121 on a paper where other children have done less well on ? i.e. so it's more about how many marks you get relative to everyone else?
Each of the two papers is marked and standardised completely separately, so if a child has completely flunked one paper, but had a score of 70/80 on the other, that paper should hopefully deliver the magic number of 121. Both papers are supposed to be of equal difficulty, so the situation you describe probably doesn't arise across the whole cohort - I suspect, although I don't know, that the number qualifying on each paper is pretty well the same.
Am I right that your DC had a bit of a drama with one of the tests, but not the other?