The law states that all schools must have a daily act of worship that is 'wholly or mainly of a Christian character'.
This is a useful summary.
I have taught in faith schools and non-faith schools, and would say that the Christian character of assemblies / collective worship is not closely correlated to whether a school is a faith school or not.
I have taught in standard state schools where local churches come in each week to act out bible stories, where all assemblies end in a prayer and where 'Christianity' is an overt part of pretty much every aspect of school life, including church visits, even grace before lunch.
Equally, I have taught in CofE schools where assemblies are moral, but not overtly religious, and where Christianity has no particular profile within the school except during explicit RE lessons.
However it is worth bearing in mind that having Christianity-based assemblies is actually the law, so unless schools break the law (which some schools do, though it is not remotely predictable whether a school will do so or not - often very dependent on the individual head), they cannot provide the 'non religious' experience that you are looking for.
Application, to see which school you are allocated, followed by deferral to January 2019 and remaining on as many waiting lists as Lambeth allows, looks like a possible way forward.