Let's assume somebody gets there first with Padme.
Qui-Gon and Obi-Wan are still there on Naboo, the three of them still end up on Tatooine, Anakin still becomes a Jedi.
Episode Two, he has that moment of "I haven't seen her in years, what if she doesn't remember me?" and finds out somebody got there first. Now he's surly about that.
Anakin doesn't get assigned to guard her; other-person guards her at the lake country and that whole thing plays out the same but with a different person.
Palpatine already has his hooks in Anakin, and -- knowing how he felt about her since he was a kid -- sows seeds of "she should've been yours."
Barring outside interference, Shmi still dies, Anakin still goes berserk on the Tuskans, but now has nobody's shoulder to cry on... except maybe Palpatine, who reassures him in all the wrong ways.
Episode Three plays out mostly as-written, up to a point.
Anakin still falls because Palpatine has been influencing him for years, the Jedi Temple still gets sacked...
...but now there's no Padme to follow him to Mustafar, and thus nobody for Obi-Wan to stow away with. The duel on Mustafar doesn't happen, and Anakin leaves Mustafar whole.
Luke and Leia aren't born, Padme doesn't die in childbirth.
Obi-Wan is still suffering from depression and PTSD but has no reason to bum around on Tatooine for the next 20 years. He tries to lay low but, much like Ahsoka, keeps getting found by Inquisitors. Eventually he stops putting civilians in harm's way and joins the Rebellion proper.
The Rise of the Empire happens as-written.
Padme continues to serve in the Imperial Senate, and she's the one who gets the Death Star plans after the events of Rogue One.
Vader -- not in armor at this point -- tracks her down. "You're a rebel and a traitor!" to old-Padme, not young-Leia.
He takes the Death Star to Naboo, not Alderaan. Despite also being from there, Palpatine doesn't actually care about the place at all and doesn't stop him from blowing it up.
Padme's husband (rather than Leia's brother) rescues her from the Death Star.
The Battle of Yavin happens. Luke "using the Force" in that pivotal moment was a huge moment of character growth for him, but not the only way to make the shot. He already said he could've done it ordinarily; "we used to bullseye womp-rats from a speeder and they're not much bigger."
Without Luke in the picture, Vader doesn't sense him, doesn't go out in his TIE Fighter during the battle.
Biggs Darklighter doesn't get shot down; he takes the shot.
The first Death Star is destroyed, with Vader aboard. Vader survives, but horribly injured, ending up in his iconic armor nineteen years delayed.
Without Leia in the picture, Han Solo never gets involved in the Rebellion. Most of the events of Empire Strikes Back just plain don't happen; there's nobody to go to Bespin, no big reveal to make there, nobody for Yoda to train on Dagobah, nobody for Vader to freeze in carbonite, etc.
The events of Return of the Jedi are an unmitigated disaster. The Second Death Star is, in fact, fully armed and operational. It's a trap, and now there's no Han Solo to infiltrate the ground base, and no Luke to turn Anakin back to the Light. Admiral Ackbar gets his ass handed to him in space, the ground crew walk into an ambush, and Ackbar is forced to flee with the rebel fleet in shambles.
Thrawn arrives on the scene, not to Sate Pestage and Ysanne Isard's Imperial Remnant, but the full might of the Empire. He doesn't need the stuff he stashed at Mount Tantiss because the Empire isn't in ruins.
(As my wife just pointed out) the Force is still seriously out of balance at this point. Would other Force-sensitive children be born to balance it? A new Chosen One?
What else happens? What else changes? What did I miss? Assume EU continuity not MouseWars.
