How can I get a job at a pediatric hospital with only one year of experience adult tele and being ineligible to apply to nurse residency programs?

Just apply! Mostly the same skills, just different interactions. Quite a few nurses on my unit did adults first and they’re doing fine now.

Personally, I stand by the 'old school' recommendation that a nurse out of school needs to do a full five years or more on a med-surg (adult) unit before transferring to a specialty unit. I have been a nurse over 26 years. I did five years on a med-surg unit before transferring to a medical oncology unit, which 10 years later transformed into an Oncology Progressive Care Unit. The nurses who didn't want to upgrade their skills stayed behind while the Oncology PCU nurses moved to a newly constructed building on campus. I have two specialty certifications--OCN & PCCN. If you are determined to skip the five years gaining experience on a med-surg unit, get fully certified in Pediatric Nursing and keep up with your PALS every two years.