Is there an English verb meaning "To become aware after surgery, to wake up from anesthesia (sedation)"?
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."