Is there an English verb meaning "To become aware after surgery, to wake up from anesthesia (sedation)"?
The phrasal verb come to, meaning to regain consciousness, is what immediately springs to mind. A doctor might say to a patient, "You will come to about fifteen minutes after the surgery is complete."
I've been an OR nurse for 15 years. We generally say the patient is "waking up," as if from a nap. There is a verb for putting someone under anesthesia, we call that "induction" but there is no specific word for waking up.
The term we use in anesthesia is "emerge" or "emergence".
A general anesthetic is usually divided into three phases: induction, maintenance and emergence.
As a means of demonstrating how we use the term, here are a few examples. "The patient had a smooth emergence." or "His blood pressure didn't become a problem until the emergence." or "He became unstable as he started to emerge from the anesthesia." or "I will give him the reversal just before he begins to emerge."