Snape might initially be shocked. He's not one to show much emotion openly, but internally he could be very disturbed. He might start investigating what happened to Hermione immediately, using his skills in potions and magic to try and find a cure or at least understand the cause.
Snape could initially be skeptical. Given their long - standing enmity, he might not believe that Harry's apology is sincere at first. He might respond with a cold glare or a sarcastic comment, like 'Potter, your apologies are as empty as your head.'
Snape might initially be taken aback. He's used to students being intimidated by him, so Hermione's outburst could shock him into momentary silence. Then he might retaliate with his usual sarcasm, trying to put her in her place.
Snape might be initially in shock. He had trusted Lily deeply, so the betrayal would hit him hard. He could become very angry and bitter, perhaps vowing revenge against her. He might also close himself off emotionally, not wanting to trust anyone again.
In some fanfictions, Snape might use his advanced knowledge of potions to brew a special healing elixir for Harry. He could also use his magic skills, like a complex spell specifically designed to mend Harry's wounds, whether they are physical or emotional.
In such fanfictions, Lily's forgiveness can make Snape seem more redeemable. It softens his otherwise prickly and often dark character. His actions might be seen in a new light, as if he is now more worthy of understanding and compassion because Lily, who he cared about deeply, has forgiven him.
Snape's character can develop in many ways. For example, he might become more open and less bitter. Surviving could make him realize that he has a second chance at life and he should use it to be a better person.
Often, Snape's Animagus form in fanfiction gives him a new sense of freedom. Let's say his form is a deer. This form could make him more in touch with his gentle side, which he has long suppressed. As a deer, he might interact with the forest animals and gain a new understanding of life outside of the dark arts and the conflicts of the wizarding world. This could gradually soften his character, making him less bitter and more open to new relationships and ideas.
Snape's character in 'Snape in love' fanfics can transform significantly. He might start to open up more, sharing parts of himself that he has long kept hidden. Love can make him more empathetic towards others, especially the object of his affection. He may also become more willing to take risks, whether it's in terms of his personal relationships or his standing in the wizarding world.
He is often very suspicious. He might immediately think it's some kind of trick by the Marauders or their descendants. He'll likely try to figure out how to use it to his advantage against those he dislikes.