In my experience Byker Hill tends to get sung to one of two tunes. I'm not altogether sure of the provenance of either, just that this is the way people sing it on Tyneside. I sing it to both, depending on which of my two zithers I'm using:

Byker Hill - Citera Alpha

Byker Hill - Citera Beta

The first tune is said to be the oldest of the two, close to that sung by the Young Tradition, though I've never heard of the shape note connection mentioned above. The second tune is softer, and may have something to do with George Welsh..