Again, the current achievements that are related to difficulty will be gone. If you already have them, they will become feats of strength and will be in that section of the achievement menu. Otherwise, it will be impossible to get them anymore.
There will likely be new achievements related to the new difficulty modes, but I don't think there has been any details on that yet.
Do they have to replay the game on Hard, Expert and Master? Or maybe they'll automatically get Hard, Expert and Master achievements?
The new difficulties are completely unrelated to the old ones. You don't have to play through the game multiple times anymore, and you can change to any difficulty that you've unlocked (by character level, not story progress) at any time, for any character.