It could be a very complex relationship. Harry has always had a deep - seated hatred for Snape due to Snape's treatment of him in the past. But with Dumbledore's action, Harry might be forced to see Snape in a new light. At first, it could be full of suspicion and wariness on Harry's part.
One possible plot development could be that Harry initially struggles with having Snape under his 'command' in some way. Since they had a very strained relationship in the original series, Harry might be hesitant to trust Snape even if Dumbledore has given him this new role. He could start by trying to figure out Snape's true intentions.
They would have a complex relationship. Snape would be a tough mentor, and Harry would likely resent his strictness at times. But Snape's knowledge and skills would also be something Harry would admire. There would be moments of tension, but also moments of connection, like when they work together on a difficult spell or potion. Over time, they could form a bond based on mutual growth and understanding.
Another possibility is that the main antagonist could be some sort of ancient evil that is awakened by the events in the fanfic. Dumbledore might have given Snape to Harry because he sensed this new threat. This ancient evil could have powers that are unknown to the modern wizarding world and it would be up to Harry and Snape to figure out how to defeat it.
The relationship would be complex. Snape is a complex character. He would likely struggle between his old grudges and his newfound role as Harry's father figure. He might be distant at times, but also show moments of great tenderness. For example, he might teach Harry advanced potions techniques in the dead of night, and then pretend it was no big deal the next day. Their relationship would be a slow - burning transformation from animosity to something like family.
One possible plot development could be that Dumbledore gives Snape to Harry as a sort of guide in the wizarding world. Snape, with all his knowledge and experience, could show Harry things that Dumbledore couldn't. This might include some dark arts knowledge that could be used against Voldemort. It could also be a test for both of them. For Snape, to see if he can truly be selfless in helping Harry, and for Harry, to learn to trust Snape despite their past differences.
At first, Harry might be very suspicious of Snape's kindness. After all, in the original story, they had a rather tense relationship. But as Snape continues to be kind, Harry starts to see him in a new light. He begins to trust Snape more and more.
They might have a sibling - like relationship. Since they are all living under Snape's care, they would grow up together. Snape's kids could be protective of Harry, especially if they know about his past and the challenges he has faced. They might share their toys and secrets with Harry, just like real siblings do.
Snape might be given a warning by Dumbledore at first. Just a simple 'stop this behavior or else' kind of warning.
In the fanfiction, Snape could protect Harry from Dumbledore by using his skills in potions. For example, he might create a potion that hides Harry's magical signature from Dumbledore's detection. Or he could use his knowledge of the Dark Arts to counter any spells or charms that Dumbledore might use to keep tabs on Harry. Snape is very intelligent and resourceful, so he could find various ways to physically and magically shield Harry from Dumbledore.
They probably have a very close relationship. Dumbledore would be like a father figure to Harry, guiding him through the wizarding world.