Carol Thatcher was a reporter on the One Show, the green room is where the One Show entertains its guests. So, when Carol Thatcher made the golliwog remark she was at her place of work: making her behaviour there her employer’s business. It is perfectly reasonable for the show to decide that someone who uses such language at work is not fit to work there.
There is a tendency of some on the right to automatically take Thatcher’s side because a Thatcher is involved in a fight with the BBC. I’m sure that there were people at the Corporation who were gunning for her because of who her mother is. But that doesn’t make using derogatory language acceptable. And before this is chalked up as ‘political correctness gone mad’, we should remember that not using language that creates unnecessary offence and hurt is just good manners.