When the Potters are hiding and Peter is the secret keeper, James Potter pulls off a Maurader style trick trap for Voldemort. When Voldemort comes after Harry, James is able to delay Voldemort enough that Lily gets the chance to flee with baby Harry. James still ends up dead.

Neither spirit-y Voldemort, nor the rest of the wizarding world, has any idea where Lily has gone with baby Harry. Snape knows Lily enough to have an idea of where she might have gone, but it's still a few years before he tracks her down again. By this time, Harry is 4 years old. Snape is with Lily for a while. Snape keeps referring to Harry as "James's brat" and Lily gets mad. "He's my son too! 'Lily's brat', Severus. Say it! Lily's brat!" She makes Severus see Harry as her boy, too. Later Snape comes and goes, always taking precautions to make sure none of the other Death Eaters know where he's gone.

Somehow Voldie finally manages to find Lily and Harry. He shows up and the story goes in a pretty predictable fashion, as it originally did in the book. Voldemort kills Lily because she wouldn't stand aside, and thanks to the blood protection Lily put on Harry the AK rebounds, Harry is alive with the Horcrux scar on his forehead, and Voldie's body is destroyed. Snape gets there in time to pull a crying Harry with a bloody forehead from the ruined house. Harry is 5 years old now.

Harry still goes to his aunt and uncle's place. However Snape comes by and threatens Petunia and Vernon. So Harry either never rooms in the cupboard under the stairs (or just spends a few days there) instead he's in the smallest bedroom. The Dursleys still ignore Harry's birthday, but the worst of the mental abuse and starvation is cut out. Instead they generally ignore him, sending him to his room to avoid dealing with him. They are worried that if they over-chore him or underfeed him, the Grim Spectre of Angry Snape will descend on their happy perfectly normal home. And wouldn't that scare most anyone?

Snape gives Harry presents on his birthday and christmas, always books. But Harry is so happy to have something new, he doesn't care when Dudley teases that "it's just a book." Imagine Dudley boasting about his Christmas toys, when all Harry got was a couple of stupid books. But the books are wizarding books, with wonderful pictures. The only people Harry really interacts with are the teachers at his elementary/grade school and Professor Snape. Harry has been exchanging letters with Snape. Harry gets an 11th day birthday present (another book) from Snape, along with a his Hogwarts letter. Harry hides the letter under his shirt, as he'd been warned by Snape about the letter.

Imagine a Snape who has come to terms with Harry as Lily's son. Or a Harry who trusts Snape. How would this affect canon?