As we come up to the third book, Creager and I must readdress the issue of the Longbottom secret-keeper. If anyone from the Order would be selected, chances are it would be Dumbledore, who offered his services to the Potters. But considering Voldemort somehow found the Longbottom hiding place, the only conclusion is that their secret-keeper blabbed to Voldemort. No one in the Order but Wormtail would be as cowardly as to do that. For the purposes of storytelling, the Longbottoms assigned Wormtail as their secret-keeper.
With no confusion as to the secret-keeper, Peter would be imprisoned in Askaban. Unfortunately being a rat, he could easily slip away. If Pettigrew decides to go into hiding as a rat and stay out of Neville's life, then Prisoner of Askaban ends up being a very uneventful book. Hermione and any number of her male friends would, I assume, do their best to save Buckbeak, but would otherwise stay out of trouble. Neville would use more Paint than Thinner, and would be followed by two blue orbs.
Pettigrew may have escaped Askaban very shortly after being imprisoned. This would allow him time to get himself into the Weasley family to get into Hogwarts. The Weasleys had a long legend of Gryffindors, so their son of the same age as Neville would be an obvious choice for getting close to the boy. Wormtail could not approach Neville directly, for his form would be recognized by Prongs and family friend, Padfoot.
Wormtail would undoubtedly have escaped Askaban long before Neville starts school, meaning the issue with dementors would have long died down. If there was any concern by the authorities with Wormtail harming Neville, there would have been a boost in security his first year, not his third. Neville, Harry, Ron, and Hermione's third year at Hogwarts is dementor free.
Creager and I mention this now, but if this were to happen at all, it would have taken place in the first book. The name Peter Pettigrew would be infamous in the wizarding society for breaking out of Askaban. When Ron brings Wormtail to school for his first year, Fred and George Weasley would spot the infamous name on their Marauder's Map. Sacrificing their map, the twins would show the name to a teacher they trust, most likely McGonagall or Lupin. Either of the two would recognize an animagus, and Wormtail would be apprehended. Peter would never flee to find Voldemort, who in turn would never recruit Barty Crouch Jr. The students would be taught by the real Moody, and Voldemort would never return to flesh form. As lovely as this would be, it is unlikely that Pettigrew could not escape from McGonagall/Lupin's confinement. Thus the fourth book begins.