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.
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 find peace. After all the chaos and war in England, they might just want to get away from it all and start fresh somewhere else.
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 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.
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 possible reason could be that the fanfiction deviated too much from the original character traits of Hermione. Fanfiction often takes creative liberties, and if it strays far from what makes Hermione who she is in the original series, it might make her seem out of character, leading to a decision to leave.
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 common reason is differences in their future plans. For example, in some fanfictions, Ron might want a simple life in the wizarding world while Hermione has grander ambitions that involve Muggle - wizard relations.
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.