Perhaps Loki is forced to stay in Jotunheim. There could be a powerful spell or curse that binds him to the realm. This could lead to an interesting plot where Thor has to find a way to break the spell and rescue Loki. It would also give Loki a chance to explore his identity while being stuck there.
Thor's perception of Loki would change drastically. Instead of seeing him as his mischievous but ultimately Asgardian brother, he would start to view Loki as an outsider, an enemy perhaps. This change in their relationship would affect Thor's self - confidence. He might become more hesitant in his actions, constantly wondering if he could have done something different to keep Loki in Asgard. Also, he would have to face the political implications within Asgard, as some might see Loki's stay in Jotunheim as a threat and put pressure on Thor to take action.
Well, Loki staying in Jotunheim could lead to a whole new story arc. He might end up ruling the Jotuns, using his wits and magic. This could potentially make him a more powerful adversary to Asgard. Thor, on the other hand, would be in a tough position. He'd probably be sent to Jotunheim on diplomatic missions or even in a war - like situation. Their relationship would be filled with more tension and misunderstanding, but at the same time, there could be an underlying longing for their old bond.
In a fanfiction where Loki doesn't leave Jotunheim, it could lead to a very different dynamic between him and Thor. Maybe Loki would start to fully embrace his Jotun heritage. He could form alliances with the Frost Giants that would change the power balance in the Nine Realms. Thor, on the other hand, might be faced with a difficult choice of whether to side with Loki or Asgard. This could also mean that Loki would not go through some of the traumas that led to his more villainous actions in the original story. He might find a sense of belonging on Jotunheim and work towards a new future for the realm.
One consequence could be a shift in alliances. Without Loki leaving Jotunheim, the relationship between Asgard and Jotunheim would be far more complex. The Asgardians might view Loki as a traitor, which could lead to Thor being caught in the middle. He loves his brother but also has loyalty to his home. Loki, on Jotunheim, could start to build a new identity for himself. He might use his knowledge of Asgard to strengthen Jotunheim, making it a more formidable force in the Nine Realms. This could set off a chain of events that would change the entire power structure of the realms.
A plot development could be Loki forming an alliance with the Jotuns against a common enemy. This enemy could be a threat to both Jotunheim and Asgard. Loki, being in Jotunheim, has the opportunity to broker this alliance, which would surprise Thor and the other Asgardians when they find out.