Really it's all speculation, isn't it? Nobody would ever openly admit to this.
I personally had a weird experience with applications that my school told me was because of this prejudice, but I take it with a pinch of salt because really... where's the evidence? They can't know, so I can't believe them.
The main point about private versus state is that at private schools you get more individual coaching, a better working environment in terms of getting your A Levels or equivalent, and often some interview practice that isn't present at state school. You're actually receiving all of these benefits, so saying that you went to a state school before is fairly moot. That's like saying you had inferior nappies as a baby, really the only thing that matters is what affects your application, in which case you're now in a 'privileged' environment, so to speak. Having previously not been at private school doesn't, I assume, mean that your teachers have therefore continued to treat you like a state school pupil. You've got the same leg up as everybody else at private school at this point.
The only influence that state vs private has is to do with these things out of your control that can cause you to under-represent your actual abilities either academically or at interview. It's not meant to be positive discrimination, it's meant to be a kind of considered compensation. The fact you're on a bursary is irrelevant. What do you want - for the selection panel to shed a tear for you? Unless being on a bursary is somehow causing you to continue to be academically under-privileged, which sounds slightly mad to me, it's not relevant to anybody except for student finance.
Where you did your GCSEs etc. should show up on your application.