> I don't think the UK is relentlessly classistLaughs in aristocracy and Oxbridge
> having an accent that is too posh in Britain is also a disadvantage
If you want to be a football pundit, maybe. It definitely won't be in any managerial / upper-level position, or anywhere where real money is around (finance etc).
> she was fair game being from a privileged background
That's the thing - she was playing the class game wrong, if she was in a setting full of people from a different background. She was not in the right job for her class. Classism works both ways; it's still classism.
> Having an accent from a different region can be a disadvantage just because
That's the natural state of thing everywhere, if anything it's less pronounced in Britain because actual language differences are minor (compared to, say, continental Europe). This is not what people are anxious about; they are anxious about revealing they hail from a region that is outside the South-East, and hence is considered poor - with all that it entails. Not a biggie in a country where bootstrapping is lauded, but a serious problem in a classist society.
Btw, I'm not British, but I've lived on these cursed islands for enough decades to experience "the system" in all its gory details.