I know this is old but I may have found a solution. I had the same problem and coudn’t import the textures in unreal engine. What I did is
- Pack the Image/Texture as embedded data: find your texture in the Properties Window. Go to Image Tab and under the Source you will se a path to the image and a picture of a box on the right side of the path. Click the box so it apears as there is a paper falling in it
- When exporting the .fbx file set the following options : Smoothing = Face, Path Mode = Copy, check the Embed Textures checkbox
- Remember to tell blender to import textures and materials. Click import all and everything should be ok