In the hospital that my friend works in working in the emergency room I asked her about that one and she said in her hospital that she works at she has never in the 15 years she's worked there heard any doctor or nurse say Stat.
They usually just say they need something immediately or get what they need immediately themselves.
However in some hospitals they might say stat but most commonly stat is used mostly in the movies and TV shows about doctors.
Saying stat means immediately so if a doctor or nurse or surgeon says stat they are meaning they need something or need something done immediately and it cannot wait.
I have read online before that some doctors do say stat so it is possible but not all doctors or surgeons, nurses etc say stat.