Final answer:
The priority nursing intervention is to relieve the pressure on the umbilical cord by placing the mother in a knee-chest position.
Step-by-step explanation:
The priority nursing intervention when a nurse notes extrusion of the umbilical cord at the vaginal introitus in a client who is in active labor with ruptured membranes (ROM) is to relieve the pressure on the cord to maintain the blood flow to the baby. This can be done through a technique called a 'knee-chest' position, where the mother positions herself on her hands and knees, with her buttocks elevated and her head and shoulders lowered. This position helps to take the pressure off the cord and prevent any detrimental effects on the baby.