I guess it must stand for Urban District, but I'm not sure. Urban Districts were usually small towns, while Cardiff has had city status since 1905.
You're right on Grangetown - Moordale Road is there, although Cowbridge Road (depending on where in a very long road it is) is either Canton or Ely.
But if your mum was in the same school as Shirley Bassey, that suggests she lived in Splott, the other side of Butetown ("Tiger Bay") from Grangetown. Despite the legend of Bassey the girl from Tiger Bay, she left there when she was two and was raised in Splott (I assume that name was not quite as romantic-sounding!).
I believe she went to school at Moorland and later at Splott Secondary: are you sure it wasn't Moorland Road your mother lived?
Gareth