We have a nursery and a nursery nurse on staff. We mostly room in, but a few moms will choose to send the babies to the nursery overnight for a few hours. We still practice couplet care. The nursery nurse will help out with assessments or feeds if we are busy, but she is mostly responsible for keeping an eye on the babies in the nursery (ie, making sure they are still alive) and admitting new babies. Any "border" babies (mom is discharge and we are just caring for the baby) are the nursery nurses responsibility.
It comes in handy for the situations you described. If mom is on another unit (usually ICU or Stepdown), we keep baby in the nursery and mom can pump and send milk until she is discharged. I've had a few exceptions where they let us bring the baby to ICU to breastfeed, but an OB nurse has to be there and bring the baby back down when it's done feeding. I don't like bringing newborns up to the ICU though, around critically ill adults who likely have something contagious. We do allow dad into the nursery to bottle/finger/syringe feed, or just hold the baby for a while if he wants.
If we have a baby who has to stay, but mom is discharged, it can either stay in the nursery or if we have rooms, mom can room in with baby (but she is not a patient and we aren't responsible for her, just her baby). This happens a lot with our NAS babies, bili babies, and kids with temp issues.
It's also useful for adoptions. Usually the birth mother does not want to room in with the baby, but it has to stay for 2 nights, so we keep it in the nursery, or the adoptive parents room in, in a separate room (if we have available rooms).
I would not be happy with your hospitals arrangement. It sounds unsafe. If they are going to have a baby on peds, either the peds nurse should take care of it (it is a pediatric patient!), or they should have a postpartum nurse on the unit and that should be her assignment, no switching back and forth between floors. What if you are on peds assessing the baby and one of your moms in PP starts hemorrhaging?
I also don't like the idea of mom/baby rooming in on a non OB floor. For one, there are really sick people in PCU/ICU. Second, those nurses don't know how to take care of a newborn. If the baby crashes, by the time you are called and make it up to the unit, it could be too late. They likely aren't looking for or would be able to recognize signs of distress in a baby until it's blue and unresponsive anyway!
I would bring these safety concerns up to your manager and propose staffing the nursery. Even if they just staff it with a nurses aide (our smaller sister hospital does this sometimes), at least it would be on the same floor and the aide could call for help easily if something goes wrong. Another idea is to send these babies to the NICU and have the NICU nurses take care of them. Another hospital in my area that doesn't have a well baby nursery does this. It is a bit of a waste of resources IMO--since the babies aren't in need of intensive care, but it sounds better than having a PP nurse split her assignments between several floors.