Currently, object textures and colors do not update after initial scene loading.
Please see my attached examples.
The video is what is happening on PC, the texture on the right cube is cycling between blank and default. The color of the left cube is cycling between red and green.
On SL Mobile, nothing happens. The cubes load in with whichever texture/color they have at the time that the scene loads for the user. Then they stay that way. Forever.
These are very simple examples, but clearly there are a lot of reasons why you may want to change a texture and have it display in-world. Texture/color changes do not update whether they are changed by script or through the edit menu by a PC user. Both methods result in no change from the mobile resident's perspective.
I have also attached a picture of my temporary workaround for this issue: By changing the prim type of the object, in this example from a cube to a sphere, it forces an update in the mobile client and the texture/color are changed for the mobile user. I've implemented this in my scripts in objects designed for mobile residents to enable some texture changes, but it's not ideal because there needs to be a delay of ~1 second before switching back to the original shape or else the update isn't guaranteed.