Good point. Take the required dress for Muslim women. Depending on which Muslim you talk to, it's anything from this
to this:
Some Muslim women don't think even a hijab is required. I've heard interpretations of the verse that's cited in support of hijabs, burqas and the like, and they take it to just mean "don't dress immodestly".
If the person can wear a scarf, then I don't know why someone can't wear a cross or a Star of David- after all, it just jewelery.
Jewelry that advertises a particular religious affiliation, though. In that respect, I can see why the BBC would want their reporters not to wear them: I imagine they have an interest in appearing balanced and unbiased in their reporting of the news, and any special affiliation on the part of a journalist is potentially a source of bias, especially if it's so central to their identity that they feel the need to advertise it.
I think the same argument can be made against Muslim women wearing the hijab or other instances where particular outward signs of one's faith are required, but I think the fact that the hijab is an actual requirement to the women who wear it puts it in a different category than optional jewelry.
In one case, you're talking about the mere inconvenience of tucking your jewelry under your shirt. In the other case, if the item really is a requirement of her faith, you're talking about actually preventing the woman from reading the news altogether.
Do I think it's important for reporters to appear unbiased? Yes.
Do I think the issue is important enough to mildly inconvenience a reporter? Yes, generally.
Do I think the issue is important enough to force a reporter to resign? Not so much.
In the way that we declare which religions are "normal" and which aren't. We might decide that it's okay for a woman to wear a head scarf because it's fairly common and we're used to seeing it. Someone else of a different faith might want to wear a full headdress of some kind, and that would seem excessive to us ... simply because it's less familiar.
However, I also think that wearing a full headdress (assuming that you mean something completely covering the face) is in a different category altogether, because it would interfere in reading the news and the job at hand much more than a headscarf or piece of jewelry.