Okay I have a solution…
Last night I was playing around with my mesh (an imported .obj), trying various things - decimating down to a tiny amount of faces to eliminate the possibility that my mesh was too complex, auto sphere project UV mapping to make sure it wasn’t something weird with the original UVs, to no avail.
I then tried separating part of the mesh, a handful of polys and using the solidify modifier. Still no go.
So then I went a step further and separated just one face, and still had the same result. So it seems there’s something weird about the object itself.
How I eventually got it to work was I created a simple cube, and then joined them together with ctrl + J, selecting my mesh first and then the cube second, so that the mesh ends up being called ‘cube’ (Not sure if it matters which way round you do this.
Then I fractured, using the default settings, and found the texture on my mesh was as it should be. YEAH!
So then just delete the fractured cube and you’re good to go.