I'm worried about the little girl who doesn't take her coat off too, but surely she must take it off at school? If she didn't, wouldn't that set off alarm bells?
Yes there was definitely a theme running about this too. The pharmacists (who's giving her the drugs) DDs mentioned it and there was a whole scene about it in their kitchen with their mum.
But then with brilliant dramas such as happy valley quite often these significant things are thrown in to be a decoy and are totally mundane.
And yes - she must take it off at school!