After reading the initial question, without the details listed, my first and foremost reaction was that Anakin Skywalker NOT turning to the Dark Side would have had drastic consequences in the galaxy far, far, away.
I first thought that if Anakin had not fallen to the Dark Side, and in turn had he not become Darth Vader, then it is likely that Supreme Chancellor Palpatine would have, just the same, become Emperor after he would have ordered the clone troopers to kill the Jedi.
Two things could have happened from then on, I thought. Either Palpatine would find another Jedi or Force user to come to the Dark Side and do his bidding OR Palpatine would have just utilized the Empire that he created, lead by Grand Moff Tarkin, to wreak havoc on the galaxy and attain the power that we see in the ensuing original trilogy of episodes IV, V, and VI.
Anakin would have likely been killed by the clone troopers, along with the other Jedi. However, perhaps he would have escaped. He would have taken the pregnant Padme with him. He would have needed to get as far away from the eventual Empire as possible, so perhaps they would have gone with Yoda to Dagobah?
So Yoda, Anakin, Padme, and the twins escape to Dagobah. Knowing Anakin and his reckless ways, he wouldn't last too long there. Vengeance would be in his eyes. Yoda would try to stop him, but not by force (or THE Force). No different than how he tried to convince Luke to not go to rescue his friends at Cloud City.
Anakin would go on a mission to overthrow the Emperor. An assassination mission would be likely. Now if the Emperor hadn't found an apprentice by then, Anakin would likely end up in a throne room with the Emperor. A duel would ensure. But thanks to the Emperor's crafty ways of manipulation, he could very well turn Anakin to the Dark Side right then and there, filling him with lies about the Jedi. And in that case, we'd be right back where Revenge of the Sith ended, with Anakin now as Darth Vader, although likely without the robotic body because there would have been no duel with Obi-Wan (Unless that came later).
If the Emperor HAD found an apprentice, then Anakin would have dueled with them, mirroring Return of the Jedi, with the new apprentice in the Vader role and Anakin in the Luke role. Anakin would likely prevail, but then it would likely go back to the Emperor seducing Anakin to the Dark Side after he likely would have to embrace his anger and hate to defeat the apprentice. So we'd be back to Darth Vader and the Emperor together... as the prophecy said... later bringing balance to the Force.
But Luke and Leia would likely have grown up with Yoda and Padme. Together. Yoda would have surely trained them in the ways of the Force, so now we'd have two Skywalker children that would eventually become Jedi Knights and Masters.
Padme would be alive and well and she'd surely be a leading force in the eventual Rebellion, pitting her and her children against her fallen love.
Now for the sake of argument, let's say that Anakin didn't go after the Emperor and he stayed on Dagobah with Padme and the twins. He likely would have raised the twins himself and clashed with Yoda as far as their teachings in the ways of the Force because, again, we know Anakin.
So then two more ripples happen.
If Yoda had to confront Anakin, perhaps Yoda would have killed him in an epic duel. If so, then there would be a secret Padme would have to keep about who killed the twins' father. They would likely find out the truth and, given that they had the genes of Anakin, they wouldn't be too happy to learn that the Jedi Master who trained and helped raise them was actually the person who killed their father. At least one of them, likely Luke, would give into his anger. If Luke didn't manage to kill Yoda because of that, perhaps he would not find out until Yoda died, with Padme confessing the truth about what Yoda did.
So now Yoda would be dead and gone. The Jedi virtually wiped from the galaxy due to the Emperor's order. Who is Luke going to learn from? The revelation would likely lead him towards the Dark Side, luring the Emperor, and now we have Luke Skywalker as Darth Vader perhaps. And who knows what would have happened to Leia. Killed by her own brother? Perhaps escaping with Padme to run the Rebellion? Who knows?
-or-
If Anakin would have killed Yoda in that epic duel, then it would be the Skywalker family of Anakin, Padme, Luke, and Leia. Having killed Yoda, Anakin would likely feel the Dark Side just around the corner. Guilt or no guilt of what he did. He'd struggle with it for sure.
That struggle would trickle down to the twins, Luke and Leia. Anakin would train them poorly, leading to all three of them being vulnerable to the Dark Side, luring the Emperor to them. And since we know that with the Sith, only a master and apprentice can prevail, then the Emperor would have to choose an apprentice from the three of