My assessment of how this could occur: It ultimately starts with Shisui vs. Danzo.
In the canon timeline, after Danzo takes Shisui's right eye, Shisui flees to Itachi, gives Itachi his left eye, and then throws himself to his presumed death, never to be seen again.
In this timeline, it seems after Danzo takes his right eye, Shisui never gives his left eye to Itachi. Presumably, he defects from the village and goes rogue since Danzo would be actively hunting him.
Mirroring what Itachi did in the canon timeline, Shisui would eventually join the Akatsuki after being introduced by Orochimaru for the purpose of keeping an eye on the organization while secretly acting in Konoha's best interests. However, in this alternate timeline, Shisui believes killing Danzo is in Konoha's best interests, and he's able to use Koto-Amatsukami to immediately subjugate the entire Akatsuki to his will, taking position as de-facto leader. Obito, watching from the shadows, is probably threatened by the loss of Nagato (because of Madara's real eyes now being under Koto-Amatsukami) but likely does not want to enter open conflict against Shisui, especially since he also wants Danzo dead anyway.
With the lingering threat of Shisui's potential intervention (either alone or as a member of Akatsuki), Danzo never pulls the trigger on ordering the Uchiha Massacre, so Itachi is never forced to go rogue and Sasuke gets to grow up with the rest of his family still alive. This radically changes Sasuke's motivations and personality, but also means that Naruto's relationship with him is not as strong, as they no longer share trauma that can help cement their rivalry/friendship.
With his relationship with Sasuke weakened to being just the same as any other classmate, Naruto has fewer ties to the village itself. With all the emotional trauma being inflicted upon him that we see from the start of the series, it's understandable he'd eventually defect from the village as well.
The Akatsuki, with Shisui at its head, eventually recruits Naruto. Since Shisui is the leader, they don't necessarily need to extract Kurama for Shisui's goals, and Shisui being an Uchiha means that he can help subdue Kurama while Naruto trains to learn to control Kurama on his own. Naruto can receive similar anti-genjutsu and chakra control training from Shisui that he got from Jiraiya during the time-skip, though he notably doesn't have anybody to teach him Sage Mode in this alternate timeline. During this time, Shisui and Naruto discuss their opinions of Konoha and, based on Naruto's testimony of the entire village shunning him despite being the 4th Hokage's son and the host of Kurama, Shisui adjusts his goal from merely killing Danzo to taking down the entire village, as it seems the entire village has become corrupted by the same prejudices and suspicion-of-future-threat that once threatened the Uchiha.
With an empowered Jinchuuriki on their side, Akatsuki finally launches its attack against Konoha, whereupon Shisui steals back his right eye from Danzo's corpse. Itachi and Sasuke are both on the scene presumably as Uchiha bodyguards and part of the Konoha Police Force. Shisui is surprised to see that Itachi has somehow awakened the Mangekyou Sharingan during his absence.