I don't think it's as simple as CoE or community school. There are outstanding and dire examples of both and all levels in between.
Where I am, the local CofE is the least good school in the area.
I would narrow down an area that you could happily live in and then go and visit all the schools you have a chance of getting into.
Bear in mind that you may have a problem getting a place at the kind of school you want as any London school that is high-achieving with all the other add-ons is likely to be very over-subscribed and you will be coming up against the class-size regulations.
The best chance of getting a place is with the Reception entry in the year your child turns 5 (applications closed for Sept 2013 start about 10 days ago) when it's an open contest. After that you are relying on someone moving in order to get a place.
If you leave the private system, then the local council will find you a school - but it will be the nearest school with a space, not necessarily a school that you would either like or that is even particularly nearby.
As you are not religious you may find it harder to get into a faith school. Round here, I have zero chance of getting a place at 4/6 nearest schools as they are so oversubscribed that they can fill the places with the baptised and church attending (and often there aren't that many places for new families to start with considering the current HUGE number of siblings everywhere).
Might be worth checking if there are schools with high levels of mobility (although that often doesn't go hand-in-hand with good results) as you are more likely to get a place.
It's worth talking to schools and the councils about which ones often have places come up. You can find details on the council websites that will tell you the last distance accepted and numbers of children in a year etc which will give you an idea of how popular or otherwise a school is.