1. Had Jaime not loved Cersei and broken his celibacy as a King’s guard, Joffrey would not have been born in the first place as Cersei would have married Robert and given him one son who died (which happened in the storyline) owing to a fever. So Cersei is left with no child with Robert. She also has other illegitimate relationship with her cousin, Lancel Lannister. But that was maybe because he was an important factor in Robert’s death. Had Jaime not loved Cersei, she would not have wanted Robert to die as she loved Robert deeply at one point of time. Jaime would have killed the Mad King and titled, Kingslayer. Petyr would not risk killing Jon Arryn as he would not be able to lead Ned to believe that the Lannisters did it.
But the most important factor of all would be everyone’s favorite, Ned Stark would still be alive as the only person who wanted him dead was Joffrey and he wouldn’t even exist.
Plus, Ned would not have any doubts against the Lannisters and in the first place would not become the Hand of the King as Jon Arryn would be alive. Hence, Ned stays back at Winterfell. The Starks would have been safe, giving a happy ending to the story.
2. Had Bran not seen Jaime and Cersei together, he would not have been pushed from the tower. Hence, Bran would be safe and would travel to Winterfell along with his father and his two sisters.
Catelyn would not have captured Tyrion, he would not have met Bronn, would not be the Hand of the King, doesn’t get framed for murdering the king nor would have been across the Narrow sea and met Daenerys. He would have just drunk, read and whored away. Sansa would have married Robert’s own, true heir. Hence, her life at King’s Landing would have been a happier one. She would have neither gone to the Vale nor would have met Ramsay or married him for that fact. The Lannisters would not have much influence in that way since the Starks become close to the royal household.
3. Had Jaime and Cersei never had incest then obviously neither Joffrey, Tommen nor Myrcella would have been born. Hence, the legal heir would be the child born to Robert and Cersei who would have been betrothed over to the house of the Starks, mostly to Sansa (if it were a boy and lets just assume it is one).
Hence, the Starks would hold a higher power in court. Jon would have still left for the Wall. Ned would have taken the threat about the White walkers more seriously and would have consulted with Robert about the solution to the problem. The Tyrells would still have married off Margery to Renly and he would be alive as Stannis would not have any motive in killing Renly for the Iron throne with Robert still being alive.
Daenerys’s story would not have changed much. She still would have her three dragons and her army, but she would not have Ser Barrister nor Tyrion by her side.
She would have still attacked Westeros, but she would not have the ships she would need as Theon and Yara would not have traveled over across the Narrow sea with their father being alive and Theon the well-living heir of the Greyjoys.
Jaime and Cersei started incest even before everything in the series unfolded, hence a lot of things would be changing completely. Robert still would have many illegitimate children but he would have also had children with Cersei. The Starks would have been safe and Jaime would not have supported Cersei as he does now.