Snape's love for Lily Potter is a major driving force. His feelings for her make him determined to save her son, Harry. He sees it as a way to atone for his past mistakes and his indirect role in Lily's death. So, his actions are always tinted with his love for Lily.
Snape's relationship with Lily Potter completely shapes his actions when rescuing baby Harry. He is haunted by the fact that he couldn't save Lily, so when it comes to Harry, he's willing to risk everything. He may even have some internal monologues in the fanfic where he compares Harry's features to Lily's, strengthening his resolve to keep Harry safe and raise him as if he were his own son, in a way to honor Lily's memory.
At first, their relationship is likely to be very tense. Harry is used to seeing Snape as an enemy. But as Snape starts to mentor him, Harry begins to see Snape's knowledge and skills. Snape, on the other hand, might start to see Harry as more than just James Potter's son. As time goes on, they develop a mutual respect. Harry respects Snape's magical abilities and Snape respects Harry's determination. Eventually, they could become something like a reluctant teacher - student duo that also has elements of a father - son relationship, especially considering Snape's love for Harry's mother.
Well, it would be a lot more complex. Instead of the animosity that exists in the original story, there could be an underlying bond. Snape might be more protective of Harry from the start, but still struggle to show it in a positive way. Harry, on the other hand, might be more curious about Snape and try to understand him better.
The development could also be due to the way the author writes Snape's backstory in the fanfic. If the author delves deeper into Snape's past, his difficult childhood, and his love for Harry's mother, Harry might start to understand Snape better. This understanding leads to admiration as Harry realizes that Snape is not just the strict and seemingly cruel professor he thought he was.
Fanfiction.net is a great place to start. It has a large collection of 'Harry Potter' fanfictions, and you can easily search for ones specifically about Snape and Lily. Just type in 'Snape and Lily' in the search bar and browse through the results.
If Lily is alive, Snape's relationship with Harry would be extremely complex. Snape's love for Lily was a driving force in the original story, and with her alive, his feelings of jealousy towards Harry might be even more intense. However, he might also be more restrained in his dislike of Harry because of Lily's presence.
As the story progresses, they might start to work together more closely. Snape could be teaching Harry some advanced magic that no one else knows, and through this process, they build a bond. Harry begins to understand Snape's past better and realizes that they have more in common than he thought, like their love for Lily Potter in different ways.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.
At first, Harry might be extremely suspicious. Snape has always been cold and mean to him in the past. He would wonder what Snape's true motives are. Harry is used to being on his own and not relying on Snape for help, so this sudden rescue would be very unexpected.
It would strain their relationship even more. Snape's love for Lily was already on shaky ground due to their different paths. Seeing her with James would make him feel like she has completely chosen the other side, and he might start to resent her a bit.