I suspect a time turner is involved for different reasoning. Specifically, I suspect that Harry's wards allow time-turned copies of himself to do things while he is asleep (since future-him knows more). So, my guess is that Harry stole Hermoine's body last night, woke up to find it missing (hence, the fumbling under sheets), goes through the interrogation, goes back in time, and steals Hermoine's body from his sleeping past-self.