I am wondering more about what is not in your question. When you indicate the nurse was concerned about getting the baby out, it raises red flags to me as to how is your baby doing. Yes, they usually measure the cord gases when they suspect an issue. I understand that is not your question. Anytime you have a difficult delivery, the pH and other measurements of cord gases are important . I don't know if this is what you are getting at. If you have questions, I'll be happy to try to answer them.
I do not see any way, of course, for the decision or omission to be corrected now. Nor, can I immediately place how you or your baby may have suffered harm. You can provide the details to local counsel to see if there appears to be a cause of action.
We do not have a client/attorney relationship until you make an appointment, we discuss your case face to face, I accept a retainer, and we explictly agree to enter into representation.
Remember, the key issue in most law suits is the establishment that some harm / damage resulted from the negligent action of another who had a duty to prevent the harm. The mere fact that someone does not by itself mean that you have a 'cause of action'.
An example of this is when someone drives negligently, but no one is harmed, it is likely that they will not owe anyone for any damage.
In this case, the Hospital (because the hospital is the employer of the nurse) agreed to save cord blood per your instructions. They didn't, but unless there is some harm, there is nothing anyone can do to fix the damage. There is no place to start.
Still, there are significant differences between the laws of the different states. A local lawyer can provide you a more precise answer.
I am licensed in New Mexico and Pennsylvania, and therefore any discussion of issues related to other states must considered within that context. In addition, my comments are not intended to create a legal representation but merely to respond to the limited facts presented by the question. Any opinion herein is not meant as a precise statement of legal rights or as a recommendation of any particular course of action. A more complete legal review can be obtained through local counsel.