Final answer:
The correct course of action is to immediately take over chest compressions to ensure the continuation of high-quality CPR, which is critical for maintaining blood flow.
Step-by-step explanation:
As a member of an emergency response team in a hospital, if CPR is being administered and you notice that the person performing chest compressions is tiring, and not allowing for complete chest recoil, the appropriate action would be to immediately take over chest compressions. This ensures the continuation of high-quality compressions, which are critical for maintaining blood flow, especially to the brain, to avoid irreversible damage from lack of oxygen. Waiting for instructions or taking over leadership are not immediate priorities when effective chest compressions are compromised.