I have just returned to this thread and i see the issue is not particularly rudeness, but defiance on the part of the granddaughter. Whether or not she is safe to be left alone aged 11 I could not say, not knowing her, but her mother, having first asked her grandmother to look after her, then deliberately undermined her authority.
Then she will be astonished when this child repeatedly refuses to do as she is told, and it will all be blamed on broken relationships, hormones, lack of love, being a teenager, no Surestart, the school, etc etc instead of simply due to, as it was so inelegantly but truthfully, described, 'crap parenting.'
My sympathy for NannySue.