It’s been a while since I looked at any of this, but I never had the impression that either of Saddam’s sons had an independent power base of their own. Saddam would not have liked the idea of anyone having a power base-if I recall correctly, at least one of them was involved with the army, but was completely incompetent in that role. The Iraqi Perspective Project report can be found online and it has some very sharp things to say about Saddam’s son when it came to his military competence. People like Saddam survive because they refuse to permit anyone to develop power bases that could be turned against them.
If Saddam dies without warning, without any time to name a heir, I think Iraq would likely fall down into civil war. There would be too many senior people afraid of what would happen if their rivals gained the top spot and would try to seize it for themselves. Very few of the administration were completely neutral and those who were – Comical Ali, for example - were never seen as threats. Ali was a Shia Muslim and I think they were a couple or so who were Christians. Saddam’s strongmen were just too divided to come up with any sort of compromise and, of course, both the Shia and Kurds would want independence.
The United States had a plan - Operation Desert Crossing - for invading Iraq if Saddam fell and the country collapsed into chaos. It is unclear if that plan could be put into action in ATL - on one hand, a lot of powers would want the chaos contained as much as possible; on the other, there will be no great justification for invasion and a lot of other powers would be concerned about the United States running up in de facto control of Iraq. Iran would certainly meddle by sending weapons and troops, open or covert, into Basra; the Saudis and Turks would be inclined to interfere too. The plan for Iraqi Freedom had yet to be devised, so I suspect there would be a considerable delay between the decision to intervene and the tanks rolling over the border. The United States had a considerable amount of pre-positioned weapons and material in Kuwait and Saudi Arabia, but they will be delays in linking up the troops with their gear. There would also be a great deal less support for the war, in the absence of 9/11.
I think it is quite likely there would be a major insurgency, along the lines of the one we saw in OTL. The Shia would want their demographic right - rule of Iraq - and the Kurds would at the very least want autonomy. The Iranians would stir the pot, as would Saudi Arabia - fewer restraints in this timeline, particularly if US doesn’t get involved - Syria and Taliban Afghanistan/AQ. How it would work out is difficult to say.
Chris