At the end of Oblivion, the Empire no longer has an Emperor, and it probably won't be getting one any time soon. There aren't even any possible heirs, so far as your character knows. They're going to have to come up with some kind of new system.
Note, however, why this is. It's not because there aren't any dragonborn. No one knows of any, but all that means is that there aren't any officially. In 433 years, no way is Martin the only illegitimate heir. There have got to be at least a handful of other people who fulfill the technical--if not the legal--requirements to be Emperor.
The significant thing is: there's no way to tell. No more Amulet of Kings, no more dragonfires, no way to be sure that someone's dragonborn. It has exactly the same effect as extinguishing the line, and it's more interesting.
It ties into the basic ideas of the Elder Scrolls world. History--reality--isn't defined by what actually happened. It's defined by what people know. It's defined by what's written down. It's defined by how you interpret it.