You probably should speak with a counselor at a school you're interested in, with your undergrad transcript in hand, to see what they will accept in transfer. Most schools will give you a idea of what they will accept in transfer at a preliminary admissions interview. Every school can provide you with a listing of exactly what courses comprise its nursing program.
My experience: I have a BA in French/Linguistics; every pre-req transferred except for A&P I and II and Microbiology, so I took those by going part time the two semesters before I began my program at a CC. I had never had those sciences at a college level--only college Physics and Biology. A&P I&II and Micro are pretty standard pre-reqs for most nursing programs
--so if you haven't taken them already you will need to.
I now take a 9 credit nursing course (feels like 15 credits, easy!) every semester; after four semesters, I'll hopefully pass the NCLEX licensing exam and become an ADN-RN. I live in the boonies so I had no accelerated program options.
RE: pediatrics. To my knowledge, the average new-hire floor nurse there is an RN without specialized training. You will do at least one pediatrics rotation as part of your RN training, possibly a second if your program offers a preceptorship where you get to pick the specialty.
Best of luck!