i checked a few things about this with my missus who is a journo and took her legal exams with her company not all that long ago. she told me the royal family has no special priviledge as regards reporting and they can only stop stories through the normal legal channels.
the story to which you refer (allegations of impropriety toward prince charles) is currently embargoed after his solicitors achieved a court order banning publication of the unfounded accusations by the mail on sunday. private individuals also have the right to do the same, assuming they have the cash to get good enough briefs.
as i would understand it the reason why overseas shows would be concerned about making direct comment on this issue is that if broadcast or published in this country they would be breaking the terms of the court order. this has not stopped internet rumours doing the mill and it has brought up the thorny issue of how such broadcast can be dealt with under legal rulings.
in the event, little to do with the royal family's position, more to do with the wealth they have to buy good legal counsel.