Seeing as nowhere I've looked explicitly states that it does so, I think it's safe to say that the answer is no. Nature Power abides by the same rules as it normally does in Trial Sites. The fact that it turns to Grass-type in Coumarine Gym seems to have been a feature present in said gym, and also in Snowbelle, in which case it turns into Ice Beam. However, it also seems that it turns into Ice Beam when used on the icy terrain of the Snowpoint Gym in Gen 4. Other than the cases previously stated, Nature Power doesn't have any different type based on what Gyms or Trial Sites it is in. It may be worthy of note though that in Ultra Space it turns into Psyshock.
For a list of what turns Nature Power into what move, you can read about it on our very own Database (here)