Is it possible to start off as a pediatric oncology nurse and work your way up to a doctor?

Medical School. Regardless of your background in order to become a physician you will need to complete medical school and post-graduate training. Although having a background of being a pediatric oncology nurse will likely make some aspects of the training a bit easier there won't be any shortcuts in the training needed.
PedsOnc nurse to doc. Yes. You would need to go to medical school (4 years), complete a general pediatric residency (3 years), and then a pediatric hematology/oncology subspecialty fellowship (3 years). You would have a unique perspective starting your career as a nurse. Other less lengthy options would be to become a nurse practitioner or physician's assistance, both of which are important positions in peds onc.
No.. But you can work your way up as a pediatric oncology nurse practitioner.
Would need to go. To medical school to get a medical degree to become a doctor.
Not usually. Educational pathways for nurses and physicians are distinct with no overlap after lower division college. A pediatric oncology nurse would have to complete college courses to qualify for medical school, get in and complete medical school (4 years), complete pediatric residency (3 years) and complete pediatric hematology/oncology fellowship (3 years).