Well, it could be that Snape has been secretly keeping an eye on Harry all along. In the fanfiction world, he might have developed a new spell or potion specifically designed to counter the effects of Dementors. When he sees Harry in danger, he uses this new creation to save him. For example, he could have a potion that creates a protective shield around Harry, preventing the Dementors from getting close enough to suck out his soul.
The main characters are obviously Harry and Daphne. Harry, the well - known hero from the Harry Potter series, is the one who does the saving. Daphne is the one in need of rescue from the Dementors.
Well, in this fanfiction, it could start with Daphne being targeted by the Dementors for some specific reason. Maybe she has some information that the dark forces associated with the Dementors don't want her to spread. Harry, being the hero he is, just happens to be in the right place at the right time. He quickly conjures his Patronus, which is a stag, and it chases away the Dementors. Daphne is grateful and from that moment on, she becomes more involved in Harry's fight against the dark arts, and they form a strong alliance.
One reason could be his sense of duty. Despite his dislike for Harry on the surface, he might still feel responsible for the safety of all students at Hogwarts, including Harry. So when he sees Harry in danger from Dementors, he steps in to save him.
The Dursleys might be left in a state of shock and confusion. Since they have been so mean to Harry, they could be worried that the wizarding world will come after them for revenge. But since they are Muggles, the wizarding world may just leave them be, but they will always live in fear of the unknown.
In fanfiction, Snape saving Harry often showcases a complex side of Snape's character. It might be a redemption arc for him, as in the original series, Snape had a very ambiguous relationship with Harry. In these fanfictions, it could be due to his love for Lily that he finally decides to fully protect Harry, which is a very interesting take on his character.
One motivation could be Snape's sense of duty as a teacher. Despite his dislike for Harry in some ways, he still has an obligation to protect the students. So, when he sees Lockhart endangering Harry, he steps in.
Another possibility is that Snape uses his position as a teacher. He might threaten Lockhart with the consequences of harming a student. Given Snape's intimidating presence and his own power within the school, Lockhart would be afraid to cross him. Snape could also call for other teachers to help, making it clear that he won't let Lockhart get away with any harm to Harry.
Perhaps Snape uses magic to create a distraction at the Dursleys' place. He could make strange noises or apparitions that scare the Dursleys, and while they are in a panic, he grabs Harry and disapparates. This would be a quick and efficient way for him to rescue Harry.
In this type of fanfiction, Snape could simply intimidate Vernon. Snape has an air of mystery and a certain power about him. He might show up at Vernon's doorstep, give Vernon a cold stare, and with his reputation as a powerful wizard, Vernon might be too scared to oppose him. Snape could then just take Harry without much resistance from Vernon.