“Snape hated Neville because… » 

“…he could have been the Chosen One and thus Lily would have lived. » 

This theory is showcased by one of Pottermore’s article, “The light and shade of Severus Snape”:

However, as this is a debate, we’ll say this: since Neville could also have been the subject of the prophecy that eventually led to Lily Potter’s death, it could be that Snape hated him as much as Harry due to this. Because if the prophecy had been interpreted to have been about Neville, Lily would have lived.

…But still. It’s a stretch.

https://www.wizardingworld.com/features/a-complex-man-exploring-the-light-and-shade-of-severus-snape

And indeed, it is a stretch. Let’s see why.

1. Snape does not hate Neville

Though it’s known that the teacher-student relationship between the two is frail, since Neville is a poor potion-maker and Snape teaches by the negative, it’s clear that Snape does not loathe him. Sure, he may dislike him, given the fact Neville systematically fails and puts his peers in danger like in his very first attempt at potions, but the most Snape has done, outside of (miserably) attempting to force Neville to succeed, is to put him in detentions twice in his entire curriculum.

First, for melting 6 cauldrons in a row in Goblet of Fire – understandable:

The next two days passed without great incident, unless you counted Neville melting his sixth cauldron in Potions. Professor Snape, who seemed to have attained new levels of vindictiveness over the summer, gave Neville detention, and Neville returned from it in a state of nervous collapse, having been made to disembowel a barrel full of horned toads.

Second, for attempting to steal the Sword of Gryffindor in Deathly Hallows:

“Your tone displeases me! The girl and her friends were foolhardy in the extreme. Thieving from the headmaster!”

[…]

“Never mind that! How did Snape punish Ginny, Neville, and Luna?” asked Harry urgently.

“Professor Snape sent them into the Forbidden Forest, to do some work for the oaf, Hagrid.”

As you noticed, the first detention required Neville to prepare potion ingredients – “horned toads” that in fact do not look like Trevor. It may be disgusting to some, but it’s fair and productive. It’s a shame that Neville had a meltdown afterwards, but even in the case Snape wanted retribution, that would be motivated by Neville’s incompetence, not, as far as he’s concerned, out of hatred for what happened to Lily. Snape’s “vindictiveness” is also noted to be unusual because of fake Moody and the incoming return of Lord Voldemort, as he can sense through the Dark Mark:

“You know why Snape’s in such a foul mood, don’t you?” said Ron to Harry as they watched Hermione teaching Neville a Scouring Charm to remove the toad guts from under his fingernails.
“Yeah,” said Harry. “Moody.”

As for the second punishment, it consisted in sending a war-hardened 17 yo Neville to the Forbidden Forest with Hagrid, Luna and Ginny, a punishment so laughable that Harry remarks:

“And Snape might’ve thought that was a punishment,” said Harry, “but Ginny, Neville, and Luna probably had a good laugh with Hagrid. The Forbidden Forest . . . they’ve faced plenty worse than the Forbidden Forest, big deal!”

He felt relieved; he had been imagining horrors, the Cruciatus Curse at the very least.

At worst, Snape has chosen McGonagall’s punishment method of sending Neville to the Forbidden Forest, except that at the time, Neville was 11 yo and the detention was set for the night. His version doesn’t match McGonagall’s cruelty; in fact, he was protecting the Silver Trio from the Carrows’ Crucio. 

If he had been as vicious as snaters say, he could have tortured Neville during his time as Headmaster in his last year, just like Harry feared. He did not.

Snape also protected Neville from being strangled twice:

• “Help me!” Harry flung at Ron, managing to get an arm around Neville’s neck and dragging him backward, away from the Slytherins. […] Neville’s face was scarlet; the pressure Harry was exerting on his throat rendered him quite incomprehensible, but odd words spluttered from his mouth. […]

“Fighting, Potter, Weasley, Longbottom?” Snape said in his cold, sneering voice. “Ten points from Gryffindor. Release Longbottom, Potter, or it will be detention. Inside, all of you.”

• “[…] And Crabbe, loosen your hold a little, if Longbottom suffocates it will mean a lot of tedious paperwork, and I am afraid I shall have to mention it on your reference if ever you apply for a job.”

But sure, tell me how Snape hates Neville’s guts!

“But Trevor!” you’ll tell me, “look at how he treated Neville’s pet toad!”

As explained in the essay “Snape as a Hogwarts Teacher”, Snape’s intention was not to bully Neville but to force him to concentrate and make a good potion, and certainly not to kill his pet (which shouldn’t be brought in a Potions class in the first place but anyway). You could argue his methods were unfair, but experimenting on creatures is part of the curriculum like in Transfiguration; not to mention, animal abuse is widespread and severe compared to the Trevor scene, so even if you consider Snape’s treatment of Trevor insensitive, it would not qualify him as some sort of animal abuser in-world. It’s time to stop blowing this scene out of proportion.

Speaking of Transfiguration, notice that he’s not the only one to criticize Neville’s incompetence publicly:

“Longbottom, kindly do not reveal that you can’t even perform a simple Switching Spell in front of anyone from Durmstrang!” Professor McGonagall barked at the end of one particularly difficult lesson, during which Neville had accidentally transplanted his own ears onto a cactus.

Would you deduct that McGonagall must hate Neville’s guts? Of course not.

“And what about the Boggart? Snape became Neville’s worst fear!”

As explained in the same essay, Neville’s “worst fear” was narratively set as a mild and childish kind compared to Harry’s serious fear of the Dementors. Neville easily overcomes the Snape Boggart (with Lupin’s help sure, but also an inferior wand and a reputation of failing even under a teacher’s supervision), defeats it a second time with “determination”, and that’s it. 

Compare this with Hermione who bursts in tears and fails her Defense Against the Dark Arts exam when confronted with her McGonagall Boggart. McGonagall who, in Hermione’s first year, sent her to a mortally-endangering detention in the Forbidden Forest, and destroyed her morale and reputation by unnecessarily retrieving 150 points from Gryffindor for being out of bed once. No wonder this girl would fear her teacher.

Finally, last time I read, there wasn’t a single quote that indicates that Snape hated/loathed Neville. Again, we could deduct dislike, maybe, but no particular resentment. 

2. Snape does not hate Neville because he could have been the Chosen One

Similarly, there isn’t a single quote where Snape says he wished Neville has been the Chosen One and resents him for it. It wouldn’t make sense either because he doesn’t blame Harry for Lily’s death in the first place. He blames himself:

“Is this remorse, Severus?”

“I wish . . . I wish I were dead. . . .”

And at most, Sirius Black, whom he thought to be the Potter’s traitor:

“Like father, like son, Potter! I have just saved your neck; you should be thanking me on bended knee! You’d have died like your father, too arrogant to believe you might be mistaken in Black — now get out of the way, or I will make you. […]”

Of course, he would eventually know Voldemort and Wormtail were the true culprits.

By the way, you have to remember: many, many other humans, including Muggles, could have been a Chosen One, per the prophecy (see: The Self-Fulfilling Prophecy). If Snape realized that, then there was even less of a reason for him to hate Neville specifically.

3. Neville being the Chosen One would have changed nothing

The first reason it would have changed nothing is that if Neville had been Chosen, he would have died. Indeed, for sacrificial protection to be conjured, the sacrificed person must have been given the option to save their own life instead. It worked for Lily because Snape had asked Voldemort to spare her, leading Voldemort to ask her five times to “stand aside” before he took her life. 

But there wasn’t a Snape for the Longbottoms. No one to convince Voldemort to spare one of Neville’s parents. 

The Longbottoms finished, Voldemort might seek out the other possible “Chosen One”, Harry, and we would circle back to what happened in canon with Lily’s death, except that Neville is now dead too.

The second reason is that even in the case one of the Longbottom parents cast sacrificial protection on Neville and thus led to Voldemort’s “death”, Lily and James might have been the target of the Death Eaters who, in canon, tortured the Longbottoms to insanity. 

Do you think Snape would wish Lily to suffer the Cruciatus until she lost her mind, rather than a swift death?

I think he’d prefer the Avada for her, if it was a choice to make. 

And indeed, she could have been killed instead, even after Voldemort’s “death”. 

Conclusion

Snape’s regret had nothing to do with another “what-if” scenario such as Lily surviving to Neville’s expense. It had to do with his own failure.