It is, actually, fully turn based. After you press wake, she takes 3 in game turns to become Groggy. If you wake her while she's Groggy, she takes 2 turns to fully awaken. If you leave her alone, she goes back to Groggy, and then to Asleep, in 5 turns each.
But I guess I need to indicate it all better in-game.
Thanks for the feedback, though. Much appreciated!