@Joshua1729 - I just re-read GoF and Lily does come out first and she instructs Harry to hold on, his father is coming.
As for the rest, I, too, had assumed Pettigrew had returned to the Potter's house immediately afterward, whether with this intention or not, and retrieved Voldemort's wand and anything else he left behind, then stashed it somewhere. He likely wouldn't have wanted it to fall into MoM hands, nor to have been caught with these items.
The theory that whatever a wizard has on them, when they transform into an animagus, transforms as well makes sense, so I don't understand why Pettigrew didn't have his own wand with him in the intervening years. They say in PoA that all they found of Pettigrew was a finger, so what happened to his wand in the shootout with Black? It would've been in his hands when Black confronted him, so it seems likely it would've transformed with him. And if it didn't, unlike with Voldemort's wand at the Potter's, Pettigrew wouldn't have been able to return to the site until the MoM staff were finished investigating and handling the Muggles, and that could have been hours or days. Also, I think Pettigrew would've wanted to put as much distance between himself and that location as possible. So either the MoM found it and that detail was omitted, or Pettigrew still had it but lost it somewhere along the way after.
As for Voldemort being unlikely to let Pettigrew use his wand, I don't agree with @Joshua1728's example. The difference between GoF and DH is the difference between Voldemort weakened and Voldemort powerful. Voldemort is so weak in GoF that he can't even remove himself from a pile of robes. He likely didn't have enough magical power within himself to use the wand, and may not have been strong enough to even hold it, since Pettigrew had to do EVERYTHING for him. He might have tolerated Pettigrew using his wand if that was their only option.
In a similar vein, how did Sirius get another wand? One would expect his wand to be confiscated by the MoM when he was arrested for killing Pettigrew. Unless someone broke into the Ministry's "evidence storage" to get his wand back, he would've needed a new wand. Now, perhaps Dumbledore was able to convince Ollivander to help Sirius after it was known that Voldemort was back, but Dumbledore would've had to rely on Ollivander not giving that info up to the Ministry and, given the MoM's approach in OotP, that seems iffy. Ollivander is a retailer, subject to Ministry regulation and oversight. It would've been in his interests to stay in the Ministry's good graces.
Anyway, I've carried on far longer than I intended. But basically, I think this is a question that could be raised about anyone who was imprisoned in Azkaban (Black, the LeStranges, etc.) as well as about Voldemort and Pettigrew.