I would probably say:
"Okay, it's time to come out."
"Your timeout is over now; come on out."
I think "come out there" sounds very awkward. "Come out of there" is passable, but I see that being used when you're sharply telling someone to come out of some enclosed area, like a large box, a swimming pool, or a doghouse:
"Joey! Come out of there! Right now!"
As for at vs in (the end of the room), most timeouts are given in the corner of a room. In fact, if you look up "in the corner" on Google images, you'll see several pictures like this one:
If the child is to stand in the corner, then the preposition is indeed in:
"Go stand in the corner."
However, if the child is supposed to stand somewhere along the middle of a wall in the room, and you wanted to call that location the "end of the room," I would probably use at:
"Go stand at the end of the room."
(For some reason, that makes me imagine a room that is more rectangular than square, where the punished child is to stand at the wall with the shorter end that is furthest away from the door – like where I've put the red ‘X’ is this blueprint:
That's where I'd go stand, if you told me to stand "at the end of the room."