Adoption Help

I had a young lady grow very fond of me as her nurse during her pregnancy. She went into early labor and delivered a beautiful baby boy. I was there for delivery and I work near the hospital so I go to the NICU often to visit him and see how he’s doing. The nurses tell me she hasn’t come back after being discharged and she gave them my number as her contact number and emergency contact so when they try to call her, they are calling me. I’m taking it as a sign that she wants me to keep him. But can I? Has anyone have any stories on assuming care of an abandoned baby in the NICU? Would he be released to me? Do we need any consent to adopt paperwork does she need to relinquish her rights? I’m afraid I’ll run out of time for a home study before he gets discharged. Would they discharge him to me?