Although their appearances might differ with a new sheogorath.

And no, they probably won't change. The appearance and characteristics of daedra are not determined by who they serve. There are independent daedra out there of all stripes, though not many since braving the waters of Oblivion alone is rarely a good idea. More to the point, Jyggalag seemed to think that the Champion of Cyrodiil would "grow into his station;" that is, by mantling Sheogorath he/she has become Sheogorath in Jyggalag's absence, and the isles will change them as much as the other way around. The "new" Sheogorath will probably be indistinguishable from the old one, apart from saying that he used to be a male orc (but thought he was a female khajiit) or something.

Only probably won't. We don't understand the full power of the deadric prince's power over their minions. Sheogorath was a prime way to be rid of the CoC, if interacted with via shrine, he could reminisce on how he used to be an adventurer much like the player not long ago (in deadric prince scope of time), maybe boast of what he did. I doubt we'll enter the isles (it would be nice) in Skyrim, so we won't see the changes.