If I were you, if what youy are looking for is for your children to:
"continue to access the same level of curriculum and extra-curricular activities" as you say, I would look not only at the private schools in the area that you are interested in, but also at the state schools (and extra-curricular opportunities available in the community) to get a fair comparison.
The state: private balance of benefits varies widely across the country, so when moving between two very different places it is often worth at least a quick squizz at both sectors, just for information.
My children, as it happens, are being state educated. However, we moved when they were about the same ages as yours. Where we moved from, I was actively looking into local private schools as the only way to access an education suitable for DS in particular. So I looked at both state and private (and also at facilities for particular interests - football and dance being my DC's passions) where we were moving to.
As it happens, where I now live the local privates aren't as good as the state + community extra curricular options. That doesn't mean that the privates aren't fine - in fact they are very much on a par with the privates where I used to live - just that the state options are very substantially better and the extra-curricular opportunities available in the town and round about are great.
The are you are moving to isn't stuffed with nationally-known top flight private schools (there is nothing, anywhere, in the state sector to match the all-round quality of e.g. Winchester, so the balance is different when such schools are in the equation) so it may be that the balance is slightly different and you may find what you are looking for in state rather than private options.
(I am also interested in your wish for 3-18. IME, children really benefit from the changes in size, scopeand character of schools that occur at a transition, e.g. at 11, and it may well be that the environment that is absolutely right at 3 may be absolutely wrong at 18... )