There is a new emperor should Mede die: if the player chooses to join the Legion after completing "Hail Sithis!", Tullius' oath still includes swearing to serve the emperor, but excludes Mede's name.
That may not
necessarily reflect a change in regime; Tullius seems to be first and foremost a man of the Empire (the institution and office,) and only second a man of the Emperor (the individual,) however close the second may be. He's also a soulless (lacking any discernable appreciation for high drama or the romantic,) utilitarian, which
can lend itself to being a good leader, and is in the middle of a bitter, bloody war, far from the empire's heartlands, regardless of any potential succession-crisis in Cyrodiil.
In other words, he's a pragmatist hedging his bets.