In The Goblet of Fire, Cedric Is killed by Wormtail who was ordered to kill the spare. However, when Voldemort gets a new body, and the Priori Incantetum takes place during the duel, Cedric comes out of Voldemort's wand. My question is, why does Voldemort lend his wand to Wormtail? Voldemort would be reluctant to share his wand with Wormtail. Why does Wormtail not acquire the wand of Bertha Jorkins and use that to kill Cedric?
Pettigrew likely had the Dark Lord's wand to return it to him once he regained a proper body.
Unfortunately I couldn't find any confirmation in the books, but JKR mentioned in a chat that Wormtail was the one who returned the Dark Lord's wand to him.
Christiana: How did voldemort get his wand back after he was in was exile
J.K. Rowling: Wormtail, desperate to curry favour, salvaged it from the place it had fallen and carried it to him. I admit that would have been a bit of a feat for a rat, but they are highly intelligent creatures!
- J.K. Rowling and the Live Chat (Bloomsbury.com, July 30, 2007)
Pettigrew would have needed to bring it to the graveyard if the Dark Lord was to have it as soon as he was capable of wielding it. Since the Dark Lord would have to be immersed in the cauldron to regain a body, leaving it with him might not have been feasible, in case the potion damaged his wand somehow. Also remember, while the Dark Lord could wield a wand in his rudimentary form, he was weak and still had to be carried as his transportation. So Wormtail, knowing what was going to happen, likely kept the Dark Lord's wand until it was done, then promptly returned it.
If the Dark Lord knew he could not feasibly keep hold of his wand during his resurrection, he would likely let Wormtail hold it temporarily. After all, he did let Wormtail carry him when he had no other options, and he wouldn't want his wand to get damaged in the potion. Pettigrew probably used it because having spent years as a rat and then going into hiding, he wouldn't have his own. He would have had it anyway to return it to the Dark Lord, so it was close to access already. He could have stolen Bertha Jorkins’ wand, but depending on what type of wand she had and the circumstances it was taken under, its loyalty might have remained with her and therefore be less effective. Also, it may have been a wand unsuited for Dark magic. With such an important ritual, it's unlikely that either the Dark Lord or Wormtail would be willing to use a wand of questionable effectiveness.
In addition, though less likely or important, it's possible that the Dark Lord might have wanted his wand to have the "honor" of restoring him to his body, and striking a few blows against the opposing side.
Once he regained his form, the Dark Lord was able to retrieve his wand from the pocket of the robes Wormtail had prepared for him.
“Voldemort slipped one of those unnaturally long-fingered hands into a deep pocket, and drew out a wand.”
- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)
Pettigrew wasn't keeping it to use as his own, or borrowing it for long. As soon as the spell to recreate a body for him was complete, the Dark Lord's wand was returned to him, and back in his possession.