The applicant to offer ratio is a Black Art!
Firstly, one of the reasons some schools ask where else you're applying is because they are trying to get a feel of how many applications each candidate is making (and it has risen from 3/4 to 4/5/6).
Also, all the registrars talk to each other about their numbers, and views of number of applications per candidate. So they some idea of how many actual children there are behind the application number.
They then have to take to account their numbers in previous years and their own ratio of offer to acceptance; and whether they needed to use their waiting list at all, and if so how many.
Also they need to knock off the number of those expected to come from their junior school (if they have one) and siblings (depending on policy), leaving the number of places which they want to fill with newcomers.
And chuck it all in a cauldron, mutter incantations and come out with this year's target figures.
There does seem to be a rise in the number of candidates across London, and it's not been matched by expansion in number of private school places. But that doesn't mean you shouldn't apply. Some schools do go very much by exam score, others interview and at least claim to attach importance to those interviews.
Shirleycantbe You need to talk to the SENCO about your DD's exam difficulties and see what they can do. They may not be able to offer extra time if her EdPsych report does not recommend it, but if they have seen it they might be willing to take in into account if they do consider all (or borderline) candidates in the round. A Head's report which highlights actual (non-exam) levels of achievement would probably be helpful too.