One possible reason could be for adventure. Hermione is a very curious character. Maybe in the fanfiction, she has heard of some mysterious magical place outside England that she just has to explore. For example, there could be a hidden library full of ancient and powerful spells that is only accessible in a far - off land.
Perhaps in the fanfiction, Hermione leaves Ron because of his jealousy issues. Ron has shown bouts of jealousy in the original series, and if that continued to be a problem in the fan - created story, it could be a major factor. For example, if Hermione started to gain more recognition in the wizarding community for her intelligence and contributions, and Ron couldn't handle it well. Or it could be that new characters were introduced in the fanfiction, and Hermione developed a stronger connection or friendship with them, which made her re - evaluate her relationship with Ron.
One reason could be that she might be on a secret mission. In some fanfictions, she could be trying to find a powerful magical artifact outside the walls of Hogwarts to save the wizarding world from a new dark threat. She might believe that she can't achieve this while staying within the school's confines.
One possible reason could be to protect Teddy. Maybe there are some threats in England that Harry wants to keep Teddy away from. For example, there could be some dark wizards or dangerous magical creatures that target Teddy for some reason.
One reason could be to escape the trauma of the war. They've been through so much in England, and leaving gives them a chance to heal in a new environment.
One possible reason could be to escape the danger in England. Given the dark forces that were at play, especially Voldemort's rise, they might have felt it was safer to leave. Another reason could be for a fresh start. Maybe they wanted to explore new magical cultures and communities in other parts of the world.
Maybe she leaves because of a conflict. It could be with a teacher who doesn't support her unorthodox ways of learning or a fellow student who makes Hogwarts an unpleasant place for her. In fanfiction, there could be a lot of creative interpretations of such conflicts.
One possible reason could be a creative plot twist by the fanfic writer. Maybe they want to explore how Harry would grow and develop on his own without Hermione's support. Another reason might be to introduce new and different story arcs that involve separation, like Hermione having to leave for a dangerous mission that she can't involve Harry in.
One possible reason could be that she's frustrated with the school's bureaucracy. For example, if she constantly faces obstacles in getting approval for new study groups or advanced classes. Another reason might be a personal quest. Maybe she discovers a hidden mystery outside of Hogwarts that she can't explore while being a student there.
One possible reason could be a misunderstanding. Maybe Hermione overhears something Harry says and misinterprets it. For example, if Harry is discussing some dangerous plan with Ron and Hermione thinks he's being reckless without considering her input, she might leave in anger. Another reason could be her own personal growth. She may feel that she needs to explore her own magical abilities further on her own and that Harry's presence is somehow holding her back in that regard.
One reason could be to escape the danger and dark forces that still lurk in England even after the main battle. They might feel that leaving is the only way to start anew and be truly safe. Maybe they want to explore other magical cultures around the world and learn more about magic that is different from what they've known in England.