Both of them have jobs, don't sponge benefits off the State, spend their days getting drunk or spitting out unplanned babies. So in what way are they "Chavy"?
Both the production team and Billie Piper refer to Rose as a chav. I'm not sure why this is even being debated.
I'm just glad the show moved away from that trend. Especially in Series 4, I think the show glorified the stupidity of the companion too much. Doctor Who has always been a show about fighting ignorance, not worshipping it.