If there will still be a Sheogorath, it will be the Champion of Cyrodil... Which we're unlikely to see... But it could be possible for him to have transformed into a Daedric Prince. After all, Aedra have been known to be turned into Daedra, and at least in the case of Talos, exceptional mortals can turn into Aedra.
If Sheogorath does in fact appear in TES:V (which he undoubtedly will) he will appear very much the same, despite having been mantled by the Champion of Cyrodil. Mortals will be none the wiser, and I'm assuming that the Mad God would keep up his original (or rather, his usual) appearance to keep it that way.
But it is entirely possible that there will be no Sheogorath at all, that after a mortal took up the mantle, that the power will fade, or that the other Daedric princes will strike out against this upstart, which has just been handed his own realm of oblivion.
In the case that the COC did indeed take up the mantle while remaining mortal, then Sheogorath will be no more (unless perhaps the COC lets another mortal take his place?). As for the Daedric Princes striking out against Sheogorath, I highly doubt it. Who knows, maybe the Princes will appreciate there being Order as well as Madness.