BBC "redness" here refers to their alleged political bias.
In the UK, the Labour Party (left-wing) uses red as its party colour, and the Conservative (or Tory Party, who are right-wing) uses blue.
You may also have heard members of the communist party (who are an extreme left-wing party) being called "Reds".
As an aside, the current Mayor of London (UK), Ken Livingstone, was nicknamed "Red Ken" due to his far-left views on many subjects.
Don't ask me why Red and Blue were chosen to represent political parties at opposite ends of the spectrum, I've no idea! :D
I hope this helps,