We are SW London (though not Battersea/Clapham) and intend to send our DD to a private school, OP.
We are more laid back, by some degree, than most of the parents around here, and this includes the state school ones.
To eb fair we are practically horizontal when it comes to this sort of thing as our own personal experiences of education have led DH and I to believe that happiness and creativity are the most important things.
We would both rather die than be, or be seen to be, pushy parents.
I appreciate that your brother is having a hard time and trust me I know the kind of people you are talking about (DH grew up in that part of London specifically, though it was different back then, so we still have friends/family in those areas) and I agree that pushy parenting is an epidemic around here.
I think is is just as bad in the middle-class-dominated state schools here, though. And I think some schools (fee-paying and state) are worse than others. Surely the head/senior staff and the ethos they establish has a huge amount to do with it? eg we are not considering sending DD to any school where we think the atmosphere is 'pushy' or where we dislike the head for this, or any reason.