When exporting a file to .FBX, the model will not preserve its textures. Instead, the materials will be applied in Unity, so it's important that the material is named something recognizable.
By default in the template file, there will be pre-named materials. Palette05 should be attached to all relevant materials, so UV mapping should remain the same between all of them.
It's also worth noting that some materials might need to change dynamically in-game, such as liquids.
