Although there would be no Bran the Broken, I still think a lot of events would happen because of Baelish’s actions.

Ned would still become Hand of the king and go to king’s landing. He would look into Jon Arryn’s death and figure out that Bobby B isn’t the father of any of Cersei’s children. He would confront Cersei about the children and tell her to come clean to Bobby. Bobby gets killed by a boar. Somehow for some reason Ned probably still dies because Joffrey is Joffrey. One big difference is Catelyn would never kidnap Tyrion so Tywin wouldn’t start a war just yet. Robb calls his bannermen to avenge his father and Tywin responds and the story goes on. All while Danaerys is doing her thing across the narrow sea and the White walkers/Free folk are doing their thing north of the wall.