@PickledOnion99
Your estimates of the number of boys with scores exceeding 125 assumes that the mean is 100. That's unlikely to be true in London and especially among the boys whose families can afford independent schooling.
While there is a great deal of debate about whether the cause is genetic or environmental, the evidence is quite strong that there is a positive correlation between family socioeconomic status and measured IQ. The average income in London is considerably higher than in the UK as a whole so I'd expect that the mean IQ of all children in London is higher than 100, perhaps something like 103. This is a small difference in the mean, but it is enough to increase the number of children with high scores by about 50%, i.e., expecting more like 3600 boys than 2400 boys with scores over 125, or 800 boys rather than 500 boys with scores over 135.
On top of that the families that can afford to pay for independent schooling are clearly better off than the London average. I'd expect that the mean IQ of children from the 20% or so of London families which can afford to pay for independent schooling would be higher than the London mean, perhaps something like 108. That would imply that there could be something like 1300 boys with scores over 125 and 350 boys with scores over 135 just from families who could afford to pay for independent schooling.
I regard Westminster and SPGS as the most academically selective schools for boys and girls in London. They have approximately 120 and 110 children each year in secondary, respectively. So, in theory, they could select with a minimum score of 135. I don't happen to believe they do this, if only because I am under the impression that the results of the pre-tests are not the primary selection criterion at these schools. However, I do believe it is possible the mean scores at these schools could be 135. And I also believe that anyone whose academic achievement is consistent with a score much less than 125 is likely to have a hard time keeping up at these schools.