Jim Carroll wrote: "I wonder how Americans feel about British singers putting on mock-Southern accents"

Well, I think it's a bit silly, but no more than if I would do it myself. (I'm an American living in Germany.) I do really loathe "phony voice singing", when singers habitually affect an accent or a set of mannerisms. However, what is one supposed to do when singing a traditional ballad that was collected in a regional accent? I really prefer such versions to the smoothed-over arrangements.

So, no ultimate answer from me, I'm afraid.

The choice seems to be sing it wrong or don't sing it at all. For most people, it's not really practicable to learn to "do" accents. Some actors can, but I doubt whether this approach is appropriate for singers (just my opinion).