Why did Voldemort wait? Who knows. Perhaps he needed time to find out who the prophecy could apply to. Maybe he was busy killing other people. I have no idea.

But at any rate, we know that Snape overheard part of the prophecy, and was caught by Dumbledore's brother. That would have been in the summer of Harry's birth. Harry was born on July 31, and Voldemort murdered his parents on October 31 of the following year. Harry was 15 months old.

Dumbledore said Snape came over "before Voldemort's downfall," so it occurred in that fifteen month period. Whatever he did to prove his bona fides to Dumbledore, I don't think it was because he found out it was James and Lilly's child who was to be killed. Indeed, if Snape had any feelings for Lilly, as has been posited by others, it would work to his benefit to leave her James-less, and possibly childless. We know from Harry's flashbacks under the influence of the Dementors that Voldemort had no intention of killing Lilly. "Step aside, you silly girl." Thus if all went as planned, the object of Snape's alleged affection would have remained alive, while his nemisis would be dead. Not a bad outcome from Severus's point of view.

So, whatever it was that convinced Dumbledore, I don't believe it had anything to do with any alleged regret by Snape for having put Lilly in any danger.

Again, just random thoughts.