You can become a pediatric nurse as soon as you graduate. To do that you'd do best to have your final practicum at a children's hospital or on a pediatric ward of a community hospital so that you can prove yourself.
That's right. There really aren't any pediatric specialty training programs out there. The final practicum is the key. You'll learn a lot about how to interact with children, how children are different from adults, a lot about the illnesses that bring children into hospital and that's how the manager of the peds floor will know that you're the right fit.