OP, I was a bit like you. I didn't want my DCs taught religion in school, but I had no choice, private schooling wasn't an option - my DCs had to go to the local CofE primary school. I was pleasantly surprised. (They were not preached at like I was growing up in a Southern Baptist part of the US, and my only point of reference with regard to church run schools.) The songs they sang at assemblies were not overly religious - I believe they were specially written song that weren't overtly religious. Assemblies were basically about how to be a good person (yes, mainly Christian views, but don't all religions want us to be nice to each other?) Yes, the children bowed their heads for a little prayer, but you can opt out of that part - the Muslim children in the school did.
Collective worship is a requirement in state secondary schools. It is based on mainly Christian values - again on vague topics like bullying, being nice to each other, not stressing during exams, etc. At the end of the assembly they are asked to sit quietly and reflect on what they have just heard and how they might be able to apply that message to their own lives.
Two of my DCs went to two different private secondary schools. One was made to take RE at GCSE, one not. One had assemblies as described above. One was made to go to chapel once a week - he didn't have to take GCSE RE.
My DCs are older now and all staunchly atheist - school did not turn them into religious nutters. I believe they have well rounded views of the religions of the world. At 20, 18, 15 and 13 they can hold competent and informed conversations about all the worlds major religions, much more so than the teenagers here in the US, who no nothing about any other religion other than Christianity or what they see on the news or tv about other religions, which we all know is not the best source of education.
Just to clarify, we were living in England when my older 3 DCs were/are in school.
Maybe it would be worth your while to attend some of the primary schools assemblies to see how they do their collective worship.