It's a tough position politically. Effectively, it's impossible to say without knowing more about the personalities which compose the council. The only member we truly know is Ocato (and Fyr, but he seems to keep to his own), and even then, it's not as though we know that much about him. The crucial point is whether they attempt to keep a (relatively, for this time period) centralized system, or try to ease tensions by allowing the empire to fall into a much more decentralized realm akin to the Holy Roman Empire (obviously, the system as it stands isn't terribly centralized, with most regions having a fair amount of autonomy, but it seemed that the Emperor's word was still absolute, if only because he had his legions and legacy. As it stands, the various regions might seek to exploit the weak position of the empire and regain true autonomy.
Still, if the next game is to take place in Skyrim (It's been about a year since I've been here, so I'm uncertain as to whether things have changed), it is possible that it could feature an aspirant to the throne, a son of Shor and walker of the Hjalti's path. Ocato's position as an Altmer might allay the desire to split off for Summerset, but it would surely be upsetting to human races, both due to racial tensions as well as the sheer symbolism of it. Regardless of speculation as to whether Ocato is a secret Ayleid or not, the Ayleids were simply a sub-sect of the Altmer/Aldmer, and thus having an Altmer occupying the ruling position recreates the symbolism of old when the Ayleids ruled over human slaves, and it is a point which the humans are likely to catch on to and which the particularly ambitious will exploit
1. This would likely lead to a largely Imperial legion choosing to ignore the orders of the chancellor, turning rather to their own generals and the like to call the shots. Without the legions, the empire is in quite the fix. Thus, the window is left open for a Nedic figure to rise up in Skyrim, gather the legions around him, and remove the "Ayleid" from the throne once again.
The scenario might arise due both to this and another possible event, namely the return of the Akaviri (It might be part of why it is mentioned that the Nerevar departed for there.
Mysterious Akavir also states, "'First,' Tosh Raka says, 'is that we kill all the vampire snakes.' Then the Tiger-Dragon Emperor wants to invade Tamriel.") The Tsaesci once held sway over the Empire and have tried to invade Tamriel by way of Skyrim, so it seems reasonable that other Akaviri peoples might do much the same. With a disorganized and headless legion standing as the only defense, it would seem to be a situation where a particularly strong-willed and charismatic figure might rise up to lead the legions and people of the north to repel the Akaviri. Then, having formed an alliance between the Nords and the Imperials, they will gather their armies the retake White-Gold from the "Ayleid", giving birth to the next Empire in much the same way as the first was born.
Such an event as this would likely be the springboard for the next part of the series. I imagine the Dunmer, under the pragmatic Helseth, would maintain a wait-and-see attitude, picking a side only when it looks to suit him best, but Summerset will likely be thrown into an upset with anti-Imperial (and even just anti-human) sentiment reaching a fever pitch (effectively, "why should we submit to these barbarians?"). Thus, it might be the case that, after Skyrim, the next game will be set on Summerset, with the scenario being the sons of Shor against the sons of Auri-El, the Padomaic versus the Anuic. It could be interesting, especially since the Emperor-to-be hasn't the Numidium as his predecessor did. Indeed, as some others have stated, all Hell may break loose.
Of course, I'm not sure how the Throat will be worked into such a storyline (and obviously, such is pure and baseless speculation. If I have hit on something after a year's absence from lore, it is purely by luck). So far, each game has held the tower and its stone as key elements of the story. I suspect Skyrim will be much the same. Perhaps the giants of the mountain can be harnessed by way of the Throat, but once again, I'm far too rusty to make any proper speculations on the matter (as for the short version of this, simply: New emperor, lots of strife, probably some o' them Daedra, and a couple of towers. Par-for-the-course in Tamriel. Well, except for the towers...).
Anyhow, the possibility of a Religious government came up. Is there a "Pope" of the Imperial Church in Tamriel, or some other type of Religious council like Morrowind's Tribunal? If so, perhaps an interim Religious ruler could keep the Empire together.
From what I could see, the Emperor was, in a sense, the head of the church. This wasn't in the "Pope" sense, but rather more like in the "King of England and Church of England" sense. There may have been a religious leader at some point, but such individuals can prove to be serious rivals to any temporal lord, and thus a strong ruler would likely see to it that their position was either abolished or severely curtailed (much like we saw with Helseth and Almalexia).
As for what the books put forth in regards to succession, my assumption is that the books are to be regarded as an alternate reality such that the author can have his story without Bethesda having to constantly look over his shoulder to correct any mistakes. It's far easier to let such novels be alternate reality since the concerns are far less and the author has much greater freedom. Thus, I wouldn't take anything that the book states as fact, such as the existence of the Emperor's son. Only after the fact, if it manages to fit within the lore and/or is well-received by the fans, would it be be moved to canon.
1The caveat here being that, although Ocato is the most vocal and visual member of the council, it is still the council that rules, not Ocato alone. Unless the council appoints Ocato as regent, then he is not truly the leader. Although appointing a regent would, in our world, be the next step in such an event, and although Ocato would be the most likely candidate, this does not mean that Tamriel operates under such measures, not to mention we do not know if Ocato is strong-willed enough to rise above the council, which itself seems to be composed of fairly powerful individuals. Thus, all of my ravings may be for naught... but why waste concepts? Without being voiced in some form, they merely fade into dream, and who knows, perhaps there will be some use for them in the end...