House Tully in 297 AC

House Greyjoy in 297 AC

House Baratheon in 297 AC

House Martell in 297 AC

Additional lore and information not covered by screenshots.

Eddard took the throne instead of Robert  which should be obvious, Jamie ended up confessing why he killed King Aerys II which lead to him being pardoned by Ned reluctantly given how such an act would have made Ned very conflicted given how it violated oaths and vows but was also clearly the just and moral thing to do given who Aerys was and what he was going to do. Jamie is released from his vows and becomes his fathers heir once again, Cersei is married to Robert and her and Jamie never get to rekindle their incestuous relationship however without being forced into he kingship her and Robert have a somewhat better relationship... there was never any love between them but Cersei wasn't abused and treated like crap while Robert reigned in his whoring and actually left his wife more or less have a huge say in governing his realm as he would often be off hunting or holding feast when he wasn't in the capitol supporting Ned. Their children are all legitimate with black heir and blue eyes of House Baratheon... Robert would eventually die during a hunt accident as in canon but much sooner and without it being an assasination.

The Greyjoys would eventually rebel but not until much later, around 299 AC around the time the conflict in the books and show kicks off in canon since Eddard proves to just as strong of a king as Robert ever would be military wise but also acts early on to quickly unite the realm as best as he can with royal progresses in mirror of King Aegon I and King Jaehaerys II leading Balon to judge that he needs to wait for a more opportune time and build his fleets up more and focus on raiding in Essos and Southoryos. When they did rebel it inevitably lead to their defeat but between a long winter and naval power it took longer for them to be defeated as King Eddard focused more or protecting his coast, dealing with a Wildling invasion of the North and keeping the realm fed over the five year long period (The Others/White Walkers didn't show up this time so they must have judged it to soon to start doing their thing with the south so strong).

An earlier divergence was Renly never being born (mistake editing the game files to set up Ned as King) and an extra year between the birth of Robert and Stannis which slightly changed their dynamic, Stannis ended up as Lord of Dragonstone but as Robert stayed Lord Paramount of the Stormlands and Eddard needed a trust worthy man to rule the Targaryen's loyalist in region it isnt taken as a slight and Ned ends up friends with two Baratheon brothers.

Jon Arryn is eventually replaced as Hand of the King as his age catches up with him which leads to him having time to father a different and more healthy set of kids then his sickly canon son Robin, he eventually died of old age a bit sooner but Lysa hasn't gone mad and with the realm at peace at the time the Vale is doing well. Little Finger was executed for financial corruption and theft due to more stringent taxation observation and record keeping under the rule of King Eddard meaning no financial crisis from his actions and a lack of Robert's stupid spending and debt.

The Martell's have a fairly good relationship with the Iron Throne with King Eddard ensuring the Dornish got justice and compensation for the deaths of Princess Elia, Princess Rhaenys and Prince Aegon which included more autonomy and the heads of Gregor Clegane and Amory Lorch, Prince Doran and Prince Oberyn both ending up on the Small Council towards the end of the third century AC although Doran's own position of Master of Laws is short lived due to his ailing health leading to him retiring and Lord Paramount of Dragonstone Stannis taking over the position during the Greyjoy Uprising in 300 AC.

Tyrion Lannister eventually finds his way into studying at the Citadel with the Maesters with Jamie around at Casterly Rock to keeps the worst of his father's mistreatment off his brother, the Little Lion as he becomes known eventually decides that becoming a Maester isn't for him and goes on a tour of Essos with his uncle Gerion before eventually returning home when Tywin dies of a sudden illness in the middle of the 290's AC. Without his shitty reputation as the Imp he ends up going to King's Landing and taking a position in the court where is intellect and talent for politics leads to him eventually becoming the Hand of the King by 297 AC with the retirement of Wyman Manderly from the  position.

Jon Snow is passed off as the bastard son of Lord Paramount of Winterfel Benjen and legitimized as his heir under the name Brandon, without this as a issue Eddard and Catelyn get off to a better romantic start and have many more children then canon although this eventually takes a toll on her health and leads to her death in childbirth.

With King Eddard lonely and in and Lady Paramount Cersie having spend the last several years as regent for her son Joffrey Baratheon the two meet up and form a friendship over time leading to the Lioness of the Stormlands slowly but tastefully catching the heart of the widowed King. The two eventually marry by the end of the decade as seen in the screenshot. This would lead to tension between the new Queen Consort and the Hand of the King as well as the Master of Whispers although the Greyjoy Uprising and the onset of the Winter and Wildling Invasion.

I don't blame you for finding it other rated, I soured on the show over time outside of the first four seasons. I'm much more of a book fan although I'm fairly disillusioned as to it ever getting finished since Martin is a lazy fat fuck who writes at the speed of glacial build up lol.