But the Guild has been around for a few thousand years, and gone through multiple empires. I will be disappointed if they are not in Skyrim
The guild is gone, yes. There are now two organizations to fill similar roles, but the guild as a whole fell with the empire. Specifically, like it was said, the guild was very much entwined in politics. Look at Ocato, the ruler after the Septims. He is a politician - one of the most powerful. But he's also a Guild Battlemage. If you've ever seen the show Fullmetal Alchemist, I imagine that the Mage's Guild shares the same relationship to the government as the Alchemists do in that show. They're an acting arm of the military in a way, serving as the specialized magical forces, while on the side they do research and pursue their own interests while the government has no need for them.
The other guilds may have stayed very much alive. The Fighter's Guild never was a very government-affiliated group. In fact, they were little more than organized mercenaries, though they themselves would never compare themselves to the less reputable other examples of mercenaries in the world. They lend swords for pay, but they have a nice moral code that keeps them in higher regard than the other mercenaries out there, the ones that believe in money first, morality later. In the uncertain times following the empire's collapse, the Fighter's Guild could have made quite a hefty sum of money from people concerned about the troubling times, now that there is no empire to protect them. And the Thieve's Guild never was affiliated with any public organization in the first place. The fall of the empire means less pockets to pick, sure, but they aren't really dependent on it. Someone, somewhere, made money after the fall of the empire, and I'd be willing to bet that the guild was still there to ease the burden of their pockets.
Other Imperial organizations, I have no idea. The Imperial Cult would be a cloud of uncertainty. While the people of Cyrodiil would continue to worship the divines in their typical church organization, the cult surrounding Tiber may or may not have been overthrown by a new emperor who wishes to prove his succession over the now-eradicated Septim line. For the most part, though, this is irrelevant to the gameplay of Skyrim, where we will most certainly primarily see the religion of the Nords. I imagine there is still an Imperial Legion, but they aren't the same legion as before. They may operate the same, but under a new banner. Considering the new empire is still trying to expand, retaining an army is a requirement. I believe Mede was a legion man himself before the empire's collapse. The Dark Brotherhood, well, they'll just go on worshiping Sithis and doing his bidding regardless of the times.
Now as for Morrowind organizations, I think it's safe to say they've been all but eradicated. Maybe some of the big houses kept some level of organization through the tragedy, coming together for mutual survival and to retain their identity. But I'd like to see the Cammona Tong attempt to drive outlanders out of a land that is no longer habitable. Then again, I suppose the large flood of Argonian invaders means more outlanders than ever. And that's a lot of opportunity