I created a material and applied it to a sphere and to a mesh. The sphere looks appropriate but the mesh is wrong. The mesh is just a solid (flashing, the material is animated) color with no texture detail. How can I apply the material to the mesh properly?
This is a problem with your mesh UVs. Open the model in your 3D modeling program (or you can also check in Unreal) and look at the UVs. I bet they are all squashed and overlapping, or very small. The default sphere there has its UVs evenly spread out over the 0-1 space.