I haven't RTFT so apologies if this is covered- but there is no such thing as "secular" schools in the UK.
State schools are required to provide an act of worship that is broadly Christian in nature. (Or at least they were a few years back when we were doing primary applications.)
What this means in practice can vary widely between schools. The ethos of a state part can be very Christian and you will only know this by visiting the school and asking questions or by asking parents of DC at the school.
In fact, there is no requirement for religious schools to provide religious worship- It's assumed they do. But it's possible that a "non-demominational" school might be more religious in ethos than a neighbouring CofE school. (Not massively likely I suspect but possible. I'm sure someone on MN had an example of this a while back).
An example - where I am, there are two local infant schools, both with a good reputation and Good Ofsted reports.
One teaches the children Christian hymns to sing at assembly, gets the DC to say Grace at lunch and has close ties with the local church.
The other does none of the above (although it does have a lovely carol service in the church at Christmas). When I asked them how they dealt with the requirement for worship, they told me they teach the children about morality in general terms and that sometimes they do things like get the children to close their eyes and think about "wonder and awe".
As a committed atheist I chose the second school. The other was way too religious for us. But sounds like it might suit you OP?
Have you visited your local non-demominational school? If not I strongly suggest you do, and ask them about how they deal with religious worship and how much Christian practice happens at the school.