The scene? Tyrion’s trial by combat for the murder of Joffrey Baratheon. Oberyn Martell has channeled his long-simmering rage into deeds, not words-- this time, he has emerged from the combat untouched, and the Mountain’s mangled, grisly corpse now soaks the arena with blood, well beyond the reach of Qyburn’s zombification powers.
In what ways do events in the Game of Thrones universe play out differently up through the end of Season 6?
Welp, without Tyrion and Varys fleeing and helping her, the Slaver Bay outcome is decidedly different for Dany, isn’t it?
Also, I don’t see the Dorne political situation going the same way if the Viper is still around.
Pretty sure things Oop North aren’t really that affected…
Tyrion would have to flee King’s Landing regardless. He has no friends there. I’m sure Varys would still advice him to go East. The killing of Tywin was mostly a crime of passion and opportunity. He would most likely still be alive. With Dorne’s need for revenge satiated for the time being, Tywin would try to tighten his grip on the north. Sansa would not be successful in taking back Winterfell. None of that bullshit with the High Sparrow would be allowed to pass.
In short: Everything’s going House Lannister’s way right until Dany’s dragons appear in Westeros.
Cersei still believes that Tyrion killed her son, so she goes after him anyway. Jamie knows this, so he would provide protection, which puts him in an awkward position.
I can’t see Shae survive unless she is clever enough to get out of King’s Landing immediately, more dangerous than Tyrion is Tywin in that scenario, since she knows too much.
Tywin is politically damaged, his reputation as well. And he made Tyrion an enemy, though he might still believe that he can salvage that situation. But he is going to concentrate on the new rival for dominance, the Martells.
Oberyn has not just gained a seat at the Small Council, he has also openly shamed and overcome the Lannisters. His brother’s plan can progress in safety. The Martells will add another powder keg in King’s Landing to blow up later.
Interest in allying with House Martell should be high among other Houses, great and small. And Tyrion’s best option at that moment is a close connection to them - and that might suit Oberyn just fine.
If he chooses to seek the Martell’s protection, he is - for the moment - safe from anyone but Cersei. He might still get an offer to vanish from Varys, yet Tyrion has a high motivation to find out what actually happened at the wedding and to his wife - which will trouble the Tyrell’s and Littlefinger.
If Tyrion starts to piece the murder plot together, they need to get rid of him. Varys might well decice to help him in this endeavour since this will rekindle the civil war, weaken Westeros even more, and make Daenerys look even better in comparison to the cesspit of House politics.
Sansa’s position is even more precarious, but Littlefinger can’t use her as a bargaining chip with Roose Bolton, unless the situation in King’s Landing would have erupted already.
Littlefinger’s preoccupation with the South for some more time might mean that no army of the Vale will be in the north when they are needed.
Though without Sansa there, Jon might act less foolish - but his future is looking worse in your scenario.
OTOH, early civil war in the South might speed up Littlefinger’s plans for the north, but that will weaken Jon’s position even more - though it improves Sansa’s.
Since Tywin will survive a while longer, the Faith Militant will not (yet) be rearmed. The Sparrow will either not gain political power at all or he will do so later during the new civil war when the Houses are hit hard already.
Unless Dany absolutely needs Tyrion’s advice (which I doubt), she is better off with him in King’s Landing where he can - unwittingly - pile the powder kegs that will blow up the South far worse than it happened in the actual timeline.
She will still find an alley in House Martell, but House Tyrell and Lannister might already be in ruins.
As her husband and one of the few decent human beings in King’s Landing, Tyrion might send Sansa off to Dorne for protection, perhaps with Myrcella. Of course, that doesn’t save her from violence or use as a pawn. It also means that she doesn’t spend the time with Littlefinger and get the political education he can give her.
Sansa was already gone when Tyrion was arrested, and no one in King’s Landing knew where she went. So, he could not send her anywhere. But her whereabouts and the role she played would interest him immensely, which might be quite some incentive for him to solve the puzzle of Joffrey’s murder - apart from the question: Who framed me?
Sigh. You’re right. THIS is why I am considering rewatching the whole show before season 7. 
I think Daenerys would be in serious trouble, possibly even dead. If Tyrion is exonerated, he still has to escape King’s Landing and Cersei’s wrath, but Varys doesn’t go with him. Tyrion probably doesn’t get pointed at Daenerys, or at least doesn’t end up in Volantis at the same time as Jorah the Explorer. Jorah doesn’t capture Tyrion, so neither of them end up at Mereen. Without Jorah in the Fighting Pits during the Sons of the Harpy ambush in Season 5, Daenerys may have been killed (remember he threw the spear that saved her life).
Even if Daenerys survives, it’s only Daario that goes to find her. Even if he succeeds, there’s no Tyrion in Mereen to push things along in her absence. The confrontation with the slavers is delayed or eliminated. Even if it happens, she goes Mad King on their asses and doesn’t get a fleet out of the deal. Her reputation would also presumably suffer.
If Daenerys dies at the Fighting Pits, Theon and Yara are stuck with a stolen fleet, a murderous uncle, and no allies. Not quite sure what would have happened with them.
Dany has a telepathic or empathic connection with her dragons, especially Drogon. Without the presence of more supporters in the Pits, she might have panicked early, which could have sped up Drogon arrival. So, Tyrion and Jorah might not be necessary for her survival at that time.
But I fully agree with you when you see her walk a much darker path without Tyrion’s advice and perspective. Mereen could have easily blown up into civil war after her departure* - though this might have also stabilized the city with one side being beaten already. If Grey Worm had won the fight, Dany might still have found a city under her rule, even more firmly than we have seen but also far more war-like.