I’m not sure about White, but he stuck to the tradition fairly well, and traditionally:
Uther got axed by Saxons, at a battle with the Saxons (fought after he was poisoned by Saxon treachery). Igraine survives, retires to a nunnery, and is eventually brought out to acknowledge that Arthur is her son (not covered in White).
It wasn’t really common knowledge that Uther had a son (see below), all of Logres was in anarchy (see below), and everyone else wanted to pull the sword from the stone and become king themselves.
Merlin took Arthur from Igraine and took him to Ector, much to her dismay but with Uther’s approval. Often, this is portrayed as Merlin’s price for giving Uther his chance with Igraine. It is often portrayed as occurring before Uther’s death.
No one was in charge; the kingdom was in anarchy, divided up into many tiny lordships, mostly at war with the Saxons. There was a Saxon king who claimed rulership (although not so in fact); can’t recall the name but he was eventually killed by Arthur.
Most of your questions aren’t ever really addressed by White. Igraine is barely mentioned, although part of that is because it’s significant to the plot that Arthur himself doesn’t even know who his mother was until the end of Book II, “The Queen of Air and Darkness”. FWIW White also had serious mommy issues and seems to have preferred to avoid writing about women.
As Lightray mentioned, in many version of the legend Igraine was still alive when Arthur took the throne. She’s often said to have entered a convent or to have gone off to some magical castle somewhere. She’s sometimes encountered by knights on their quests.
Arthur’s legitimacy was questionable. I believe this is addressed briefly in “The Queen of Air and Darkness”, but Igraine was already married to Gorlois, the Duke of Cornwall, when she met Uther. Merlin helped Uther to magically disguise himself as Gorlois so he could enter Gorlois’s castle and have sex with Igraine* while Gorlois was away. Gorlois was killed in battle that same night.
When Arthur came along there was no way to prove he’d actually been fathered by Uther and not Gorlois. I don’t have a cite for this, but I’ve also read that while a child conceived before its parents were married would normally have been considered legitimate as long as the parents got married before the child was born, this was not the case if the child was conceived through adulterous (rather than merely pre-martial) sex. So if Gorlois was still alive when Arthur was conceived – and again there was no way to prove whether he was or not – Arthur could never be Uther’s legitimate son.
The whole pulling the sword from the stone thing served as magical proof that Arthur was the son and rightful heir of Uther, but prior to that his status was questionable and it would have been rather awkward having him around if Uther were to have any younger, clearly legitimate sons. Better to quietly foster him out and send for him later if needed. IIRC then in some accounts there were also attempts on young Arthur’s life so he was sent to Ector at least in part for his own protection.
*Many modern retelling have Igraine in love with Uther and fully aware of his identity, with the disguise being just to fool the guards. But according to tradition she was a faithful wife and only had sex with Uther because she really believed he was her husband.