Forget about St Joseph's the Catholic school - oversubscribed and you have to attend the church (and have the form endorsed by the priest) to get into the school.
The other top school is Wallace Fields - it is difficult to get in to - prctically have to live on the doorstep and no guarantee that those who attend the infants (5-8) will get into the junior school.
At present, St Martin's entrance criteria meeans that if your child is in the infant school they will have a place at the junior school.
Funnily enough, my closest school is Southfields, but because of the way the appeal process and admissions have gone, my daughter was allocated St Martins on the other side of town. (in spite of living less than a mile from Stanford Green our first choice).
I think you need to look at the admissions procedures and rules.
As far as nursery places - most good pre-nurseries are again oversubscribed. The young children tend to get afternoon sessions (never ideal) and no guarantee that they will get into the adjoining infant school.
As far as private schools go - there is a good pre-prep (mixed) St Christopher's, Kingswood House (boys prep school with girl's allowed up to a certain age) and nearby Ewell Castle (mixed) in Ewell. There is also Epsom Lodge - pre-prep of Downsend school.
Hope this helps. The Surrey website will have all the info about applying and salient dates