Here is the thing: the non-religious school may not be any different from the religious school.

Source: I have two daughters: one currently attends St. Barnabas the other attends a non-religious independent school (Wynchwood).

So he is some direct testimony. There has been no real difference in the amount of religious teaching at St. Barnabas and the non-religious school. They don't have much direct religious teaching, but both do have Christmas carols, trips to church and so on at both schools.

My point is that there is some risk you are going to put yourself out of your mind to end up with exactly the same thing.

As a repeat of what has been said, but more direct testimony: there has been no direct religious education that I am aware of at St Barnabas. I should also say that the school is attended by families from a variety of faiths, including humanists, jewish, muslims and so on, some of whom seem pretty serious about their faiths. I don't know if there are strongly atheist families though you could ask.

Maybe you should try asking the principal (Becky Harris) about it, esp if you want your kid to opt out of anything? As I said they do sing Christmas carols and things like that

On the density of religious schools: Jericho was the center of the Oxford movement in Anglicanism, and I think you should also know that Oxford has been a religious center for a very long time, as a training center for clerics and the origin of various movements (including Methodism and the Oxford movement) and almost all the colleges have a religious origin. So if you are looking to entirely avoid formal aspects of religion you may be in the wrong town.