You might be thinking of Radio York which did actually rebrand briefly as BBC North Yorkshire. The story goes that they did this without consulting anyone at a higher level in the BBC. Apparently the head of regions was driving in the area and wasn't best pleased when he tuned in to discover that the station name had been changed without his knowledge.
The whole station wasn’t rebranded, they launched a split breakfast show, with Radio York Breakfast covering mainly urban ‘Greater’ York and BBC North Yorkshire the more rural part of their big county patch.
I think they also used split jingles and split news bulletins for the rest of the day too
Write that down in your copybook now.