I admit I’m having trouble with the premise of this AU because I can’t see Cersei having Robert’s child. Cersei took care to rarely have penetrative sex with Robert and was quick to abort the one child they conceived together. If, say, she got pregnant after sleeping with both Robert and Jaime and wasn’t sure which fathered the baby, I reckon she’d get an abortion anyway rather than run the risk of having Robert’s kid, which she was adamant on preventing and took great pleasure in. A Cersei who had Robert’s kid wouldn’t be Cersei.

But…. assuming she got a personality transplant or, idk, was drunk and forgot she slept with Robert, I don’t think she’d have that good of a relationship with the kid. Cersei’s affection for her children is intrinsically tied with how she sees them as extensions of herself and it’s hard to do that with a black-haired blue-eyed child that looks the image of the husband she hates. Considering what she says to Ned about being “cleansed” of Robert’s child and how much joy she took in depriving Robert of biological children, I don’t imagine her being too happy with that Robert-looking child or too fond of them. Ignoring them would be the best case scenario.

However, that child would change the plot completely. With a child so apparently Robert’s, Stannis would have no reason to suspect the children’s parentage or go to Jon Arryn with his suspicions. That scraps Jon’s investigation altogether, though whether that’s enough to save his life is unknown. Jon’s poisoning was primarily triggered by his plans to take Sweetrobin from Lysa and foster him with Stannis which prompted Lysa, with encouragement from Littlefinger, to poison him to prevent that. While the fostering was partly because Jon wanted to ensure his son’s safety when he made his move against the Lannisters, he also wanted to distance Sweetrobin from Lysa’s coddling that he believed made his son weak, so the latter motive still stands even if the former is butterflied away. Note, however, that Jon did show signs of improvement when Maester Colemon purged him in OTL so there’s a chance that he might have survived the poisoning if Maester Colemon had been allowed to continue treating him and hadn’t been sent away by Grand Maester Pycelle. As Pycelle was aware of both the incest and Jon Arryn’s investigation (he is the one who provided both Jon and later Ned with the genealogy book), he assumed it was Cersei who poisoned Jon and interfered in his treatment, only providing milk of the poppy and leaving Jon to die because he believed that Cersei wanted Jon dead. So it’s possible Pycelle might not just leave Jon to die if he didn’t know that Jon posed a danger to Cersei and that’s why she needed him dead. If Jon Arryn survives and suspects his illness was actually a murder attempt, he might start looking closer at those around him and might be able to trace where the poison came from, while Ned stays in the North to deal with what’s happening at the Wall.

If Jon dies, whether because the purging didn’t save his life or because Pycelle still left him to die, say, on the assumption that it was a power play from Cersei to make way for Tywin’s Handship or something to that effect, Robert still goes north to make Ned his Hand and Lysa sends the message accusing the Lannisters of Jon’s murder to Catelyn. It’s possible that Stannis leaves King’s Landing in a huff over Robert naming Ned Hand instead of him, but it’d be far easier for Ned to get hold of him, that is if Stannis didn’t return to King’s Landing in response to Ned’s letters or out of his sense of duty. With Stannis in King’s Landing, Ned is far less vulnerable to the manipulations of Varys and Littlefinger, not to mention that there is no investigation for him to trace and no lighting bulb moment over Sansa’s comment about Joffrey not being the least bit like Robert. While everything with the dagger, Catelyn arresting Tyrion and the Lannisters attacking the Riverlands still happens, Ned probably doesn’t get ambushed by Jaime (no investigation = no talk of Jon and Stannis visiting a brother = no Littlefinger taking Ned to see Barra), there’s no confrontation with Cersei, Cersei might not even poison Robert when she did, spurred by several people investigating her and Renly plotting against her. Even if she still poisons Robert, without Ned learning of the affair and of Joffrey’s true parentage, the will naming Ned regent might result in one hell of a power struggle but Ned does not get arrested or executed and the North does not call its banners.

That’s one reason for the War of the Five Kings butterflied away. One king down. Another two don’t declare themselves king at all because without the adultery suspected and investigated, Stannis never claims the throne regardless of how little he likes Cersei or Joffrey. Renly’s plot to install Margaery as Robert’s queen never happens and he doesn’t declare himself king. That’s most of the plot causing the War of the Five Kings undone.

That doesn’t necessarily mean things would be peaceful though. Cersei isn’t letting Ned be regent that easily and Ned isn’t rolling over and letting Cersei handwave Robert’s will away (neither is Stannis for that matter, even if he grits his teeth over Robert picking Ned over him again). Don’t forget that Joffrey is still a little shit. Bran’s defenestration still happened and he is gonna remember who pushed him and why soon. Tywin Lannister has openly assaulted the Riverlands and Ned has an arrest order on Clegane. The Tullys and the Lannisters are a hair away from open war. The Starks are already suspicious of the Lannisters, and Lysa has accused them of Jon Arryn’s murder in front of the Vale’s nobility. Both Varys and Littlefinger know about the incest and are invested in inciting a war so don’t count out their machinations too. There is still plenty of reasons for conflict, albeit with fewer kings in the realm.