Now, in Morrowind there's some discussion on how the Emperor's health is failing. I've heard people say that he recovered before Oblivion began, but I don't believe it.

For one, the man is obviously not well. When you meet him, he definitely looks pale, and his eyes seem sunken. That does not look like a healthy person to me. Also, the last thing he says in the intro movie is "But I am dying...". This can't be referring to his future assassination because even though he has the idea that assassins are going after him, he couldn't possibly be sure that they WOULD kill him and he wouldn't get away. He must be referring to his health, then.

Is there any actual confirmation that Uriel's recovered? Because I'm pretty sure that the old chap'd be on his way out anyway, even if he wasn't assassinated, I think he would have died from his illness.

I was under the impression he had foreseen his death in a dream, just like he had seen the future-Champion Of Cyrodil
But he probably was going to die from his illness if the assassins left him alone but I'm sure he referred to the assassins in these messages

People knew that there would be trouble regarding the throne, what with the emperor's failing health and the accusations of his heirs being dopplegangers, but once they've all been assassinated, it is made clear that the worries of yesterday are small compared to the troubles of today. Not to mention that the previous worries ended up amounting to nothing.

Uriel Septim is 87 years old, that he looks every bit that age is pretty normal, also your average octagenarian isn't the most robust individual, and I'm sure his health was quite fragile. The intro video is all about the precognitive powers of the emperor, he knows he is about to die but not the exact blow by blow account of it, in theory he could have died of a heartattack due to the stress of trying to escape but that wouldn't be very dramatic.

Also the rumours we hear were not started by the Emperor, I'm sure he knew even when Jagar Tharn imprisoned him that he wasn't going to die just yet, but no one else did so with him being so old it's not surprising that the people were getting antsy when he gets ill, by all accounts even if his sons weren't Tharn's stooges they were pretty nasty individuals thanks to thier mother being the ultimate female dog.

He certainly does seem to suggest that he knows exactly when he will die, I am fairly certain he was referring specifically to the assassins in this case, especially given what he says when giving you the Amulet of Kings just before his death, it sounded like he knew he was going to die then, and that's why he chose that time to give you the Amulet.

Now whether he was sick at the time or not I don't know, I see nothing official that says he had recovered from his illness, but nothing that says he had not either, though he did seem to be putting up relatively well with the stress of his escape for someone in failing health. In any case, I guess it doesn't really matter, the assassins got him, and clearly he knew he was going to be dead soon one way or another.

Well, he is 87, generally that's considerd to be rather old (unless you're a mer).