The power balance in the fanfiction world could be different. Without Lucy, Fairy Tail might lack some of the strategic and magical capabilities she would have brought. At the same time, Lucy could be a major force in another part of the fictional world, changing the political and magical landscape there. It also means that the relationships and story arcs that typically involve Lucy in the original Fairy Tail story will be completely different, creating a whole new set of stories and interactions.
Well, in a fanfiction, Lucy might join Fairy Tail as a kid full of wide - eyed wonder. Maybe she stumbles upon the guild while on an adventure with her family. She could be immediately drawn to the warm and rowdy atmosphere of Fairy Tail. The members might see her potential and welcome her with open arms. She could start learning simple magic tricks from the more experienced wizards there.
Natsu could get really angry at Lucy. He has a bit of a temper, and being tied up against his will might make him lash out at her once he gets free. Another consequence could be that the other guild members might find out and either laugh about it or scold Lucy for her actions. It could also lead to some sort of competition between them, like Natsu trying to prove he can't be tied up so easily next time.
If Lucy never joined Fairy Tail, the friendships and relationships among the characters would form in a very different way. Also, many of the story arcs might not unfold as they do in the original plot.
Maybe Lucy finds a long - lost spell book that belonged to a powerful Celestial Mage in the past. The spells in this book are the key to saving everyone. She spends time deciphering the ancient spells and finally manages to use one that can nullify the threat facing Fairy Tail. It could be a spell that can erase the existence of a powerful dark magic or something like that.
Another scenario could be that in a battle against a new, extremely powerful foe, Lucy is caught in a crossfire between two massive magical attacks. Despite her friends' attempts to protect her, she gets fatally injured and dies. This would have a huge impact on the other characters in the Fairy Tail guild, leading to themes of grief, revenge, and moving on.
Perhaps Natsu is under the influence of a powerful and dark magic that makes him lose control of himself and accidentally hurt Lucy. It could also be that Natsu is trying to protect Lucy from something but his actions end up causing her harm instead. Like if he tries to keep her away from a dangerous situation by pushing her too hard and she gets injured.
It could be that Lucy leaves Fairy Tail because of a personal quest she has to undertake. This quest might be related to her family or her past, and she feels that she has to do it alone without the support or interference of the guild. She may plan to return once she has completed this important task.
She might use her celestial spirits in unexpected ways against the guild. For example, she could have her spirits create illusions to embarrass the members.
One possible plotline could be that Lucy uses her charm and intelligence to plan elaborate schemes to get Natsu's attention. For example, she might invite him on special adventures that are just for the two of them, where she can show off her skills and bond with him more deeply. Another could be that she gives Natsu special gifts that are meaningful to him, like a scarf she made herself which has some magical protection. Or perhaps she starts dressing in a way that she knows Natsu will find appealing, not in a vulgar way but in a way that highlights her beauty and confidence.
Perhaps they leave because they feel underappreciated in Fairy Tail. They might think that their skills and contributions are not being recognized enough, so they decide to go off on their own to prove their worth elsewhere. They could start their own small guild or join another existing one that values them more.