What an interesting question, Fleebee.
Almost certainly, Thatcher and Atkinson would have signed a contract which bound them to uphold the ethos and reputation of the BBC.
Given the outrage that their remarks caused - and the clearly stated position of their employers on issues of race - it would have been very very difficult for either of them to argue that they had not broken their contract.
Had they not signed such a thing, I would suggest that it would have been possible for them to have made a case for unfair dismissal citing the Universal Declaration of Human Rights in which "everyone has the right to freedom of opinion and expression; this right includes freedom to hold opinions without interference, and impart information and ideas through any media regardless of frontiers". (Atkinson less so, as his remark - though intended to be private - was transmitted therefore he was representing the BBC at the time and not himself as a private individual.)
So there is the right to freedom of speech and there are the contractual laws which you have agreed to and are bound to adhere to. It's a tricky balance, isn't it?
However, the difference between Thatcher and Moir is that Moir's opinion is a fundamental part of her job. It is her job to have an opinion. Furthermore, it is her job to have an edgy and controversial one. I would wager that Moir has no such contract.
It would strike me as terribly unjust for someone who was employed to spout controversial viewpoints to suddenly be fired because this particular one is just a wee bit too controversial, despite the fact that she has stuck (in my view) to the professional guidelines laid out for her.
We as readers enter into a tacit agreement when we buy a publication that we are largely subscribing to the ethos of that publication (in the digital age this becomes a much foggier issue of course, as we no longer have to purchase news to read it).
I have said previously that I would be stupid to be shocked by seeing images of naked breasts in a copy of Playboy. That is why I don't buy it. But while it is legal for such images to be published, I have no right to stop others from buying it just because it would offend me if I bought it.
Therefore, if you buy The Guardian you are expecting the reports and features to be of a left-wing slant. If you buy the Daily mail you expect the reports and features to reflect a largely conservative and left-wing mindset. This doesn't mean that nothing inside it can offend you or that you do not have the right to state this if it does so.
But this is why we have vast numbers of publications in this country. Because we value the right to "?hold opinions without interference, and impart information and ideas through any media?"
*I am not a lawyer, so am fully prepared to be corrected on any of the legal issues I have just commented on.