There's a difference between Public School and Independent/ Private school!
There a quite a few reasonably-priced private schools around, by which I mean �7-10k a year rather than per term
You'll find less snobbery and more children of people like yourself who aren't necessarily rich but are making the effort.
However it really depends on your child's character and their talents, whether they need the extra encouragement and supervision for example. Despite popular belief, state schools are often better funded in terms of science labs, computer equipment, music studios etc
As I am "bookish" my move from comp to private suited me fine. English became a subject of study, rather than an exercise in occupying myself with a book while the teacher tried to keep my classmates' attention.
Had I been interested in Fine Arts, Acting or Music I would have stayed at the comp as they had really very good facilities and staff (ITV once filmed and broadcast the annual play)
So, get the prospectuses, do your homework and ask your kids !
Good luck