Peter Pettigrew did not wait 14 years to retrieve Voldemort's wand from the Potters' home as the added first line of your question suggests. After retrieving the wand almost immediately, he was in hiding for 12 years (not 14 years).
"Harry Potter and the Prisoner of Azkaban":
You haven't been hiding from me for twelve years," said Black.
"You've been hiding from Voldemort's old supporters. I heard things in
Wiki has your answer:
Upon his curse rebounding upon himself, Voldemort's wand was blasted
from his body and landed not far away from his body in the ruin of the
Potter home. After Rubeus Hagrid rescued the infant Harry from the
wreckage of the house later that evening, Peter Pettigrew returned to
the house and discovered the Dark Lord's wand in the wreckage,
taking it with him as he fled the scene in order to prevent the wand
from falling into the Ministry's hands and possibly exposing him as a
traitor since all wands have a record of the spells they have cast and
can be examined by those trained to do so. He would keep this wand in
an undisclosed location for twelve long years afterwards, until the
time when he could locate its true owner once again.
Tom Riddle's wand
 Wiki has a link at the bottom in the notes that used to link to the 'rumours' section of JK Rowling's website.
Some users on reddit also verify having seen her comments on the rumours section of her website. The exact quote they attribute to JKR is:
"Strong feat for a rat, I know." http://www.reddit.com/r/harrypotter/comments/3935we/voldemorts_wand/
The rumours section of her website has been since deleted. I tried searching for the interview but could not find it.
FYI, the term "rat" here is a double entendre; a word with two meanings. In the literal sense, Peter physically could transform into a rat. However, figuratively he was a rat, somebody who snitches, which Peter did by outing the Potters. So it could be that JKR is just talking about Peter figuratively. Further when Peter transforms physically into a rat, so do his clothes, his wand, etc. So why couldn't he simply carry two wands on him in human form and then transform into a rat? Either way, JKR clearly states that Peter did get it almost immediately, and she is authoritative in canonical terms.
So your answer is in my first paragraph from wiki. Peter Pettigrew got the wand, as he didn't want anyone else discovering it and possibly exposing him as a traitor.
There is no mention of this in the book directly, but an author or writer's comments or answers to questions are canon and as good as what's in the books. Sometimes it's even better, as writers will tell you just because a character in a book says something, doesn't make it so. As explained to me by more experienced users on the site, it is commonly referred to as "The Word of God."