Personally, I say look at the Furzedown and Tooting options!
Private school is overpriced and over rated - many of the local state schools are performing exceptionally well and have a merry band of kids and parents attached to them.
In Tooting, schools of note are Furzedown Primary, Penwortham, St Boniface, Tooting Primary School. All come with excellent parent write ups and high achieving kiddie winks.