At the much less suburban end of the scale, there is Camden School for Girls.
My step-daughter left that school last year with a bunch of good A levels. She absolutely loved it - she rather misses it now, as she's found her first year at college less inspiring and challenging. It says a lot about the expectations of the school and the quality of its teaching.
Though non-selective (they ensure there are girls of all abilities in the school) it has been consistently excellent for decades and continues to produce hordes of strong, confident and academically brilliant young women each year.
The location is not for the faint hearted (Amy Winehouse territory!) but Camden girls are famous for their street wisdom and sharp wits.
I can't praise the school highly enough. The leadership is outstanding and the teaching is at least as good as that of the most prestigious private schools in London. They prepare their brightest students extremely well for admission to elite universities - in fact, their 6th form attracts a very large proportion of privately educated pupils who come to it to increase their chances of entry into Oxford and Cambridge.
What's interesting about Camden is that it has a pretty good bunch of primaries as well, so if you are interested in London 'proper', rather than the suburbs (), it's worth a look.