Thank you for your responses so far. We can definitely afford a normal private school tuition. As much as I would like to do the American Schools the location and price of those make it not much of an option. What will a private school do for us that the regular ones don't? My girls currently attend Catholic schools and have done so their whole life.
Just doing a quick search on the tuition fees for the American schools vs. the tuition fees for private schools in/near Leamington Spa, they actually seem fairly similar. For private education (whether American Schools or UK schools), you're looking at approximately £10,000 per child per year (for UK schools, it's about £3,000 per term - three terms per school year, so that's at least £9,000 in total per year and the American Schools charge between £8,000 and £15,000 per year).
I am British and was educated at a state school here in the UK (free 'public' school in US terms) - I got better grades than my friends who attended private school and a great education - I applied to Oxford University (although unfortunately was not offered a place) and went on to get an undergraduate masters degree from a good UK university (ranked in the top 20 now I believe), and then a postgraduate research masters degree from one of the top universities in the country. Earlier this year, I started a PhD programme in the US (although was not happy there and moved home in the end).
Personally, I don't feel that private schools are necessarily better than state schools. If you can find a good state school, then your kids should get as good an education as a private school would give. The main reason that private schools get such good results is because they only enter pupils into exams that they know will get an A or a B grade thereby making it look like all their pupils are amazingly intelligent! The national school curriculum is the same all over, so all schools should be teaching the same material.
I'd say the main benefit of private school would be smaller classes and possibly better behaved kids (although I know several ex-private school pupils who are incredibly snobby and stuck-up!).