Just a note to those comments that insinuate the private education is always better - that simply is not the case. True, they teach directly for the tests that provide entry to selective secondary and boarding schools, but you simply cannot make that generalization.
In fact, many studies show better performance at university level from state educated students:
http://www.theguardian.com/education/20 ... te-schools
It has more to do with social status, extracurricular perks and sport. Oh, and don't forget admission to the 'best Bonfire fireworks in SW London!'
My kids are at a top Catholic, state primary and at least 1/2 the students are from French, Spanish, Italian, etc families and English would be not always spoken at home. You'll likely find that more in a state setting than independent. At this primary, over half move on to very selective independent secondaries, so the education is definitely on par without the high price tag (amazing community to boot).
So it boils down to finding the right fit, getting in (oh yeah, there's the rub), and what matches your own family priorities. It's a great area, you will love it, but the schooling issue can be the very hardest part of coming into London with children (I have four, and came when the oldest was year 5...so hard!).
Good luck!