Believe me, children do notice things like accent at a very young age.
When I was about 9 years old, my parents let my 7 year old sister and I go to the cinema on our own for the first time. After the film had finished we waited outside for them to come and pick us up. A group of about 6 kids who must have been about 11 or 12 were walking past and one of them asked me what the time was.
I told them what the time was and next thing they were practically rolling around on the ground laughing and mimicking my accent and asking if I had tea with the Queen, where my pony was and if mummy and daddy drove a roller...
I was mortified - what on earth does a 9 year-old do confronted with a group of slightly older children who think that the way they talk is to be mocked. They had just asked the time, they weren't looking for a way to interact for whatever reason, but my accent had provoked the response it did.
My father was a rural GP and we sent to the local Brownies group - we were pretty much ostracised because we went to a private school and didn't speak the same way. After 12 weeks I was so unhappy that I begged my mother not to send me anymore. My father was shocked as he knew all the families and that they were lovely people.
I got to know some of the girls from Brownies years later when we were all working in a local restaurant - and they were very nice, and a couple said they felt bad about being so horrible back when we were children. One of them is a very good friend of mine today.
I'm not nearly so worried about DD going to a London primary school - she's at a very un-posh local nursery that I love and she is very happy at. The children are Indian, Chinese, Japanese, Middle Eastern, Black African, Black Carribean, Portugese, Russian, Polish, Lithuanian etc - such a mix that no-one is really different because everyone is. Plus 'class' is such a terribly British obsession at all levels along with a reluctance to celebrate success and achievement unless it takes place on x-factor or the football pitch...
Anyone who doesn't get the OP's point has obviously been extremely lucky to not have been bullied at school for being different.
I also think responses might have been very different if the OP had said that she was worried about sending her non-RP speaking child from a working-class background to a private school and whether he/she would be picked on for speaking in a different way.