I have the same problem. I exported my project as obj and opened it in Twinmotion (2023.2.3), which went well. I then saved it as a datasmith file and imported it in unreal engine (5.5.3) but none of the textures go through. Geometry seems fine.

I think the issue is between Twinmotion and unreal engine though, because according to the documentation Twinmotion is supposed to created a folder called "Twinmotion" under "Contents" folder when exporting the datasmith which it doesn't.

Did you do it through Twinmotion or another way?