Additional Turns and Emrakul, the Promised End
It doesn't matter who controls the turns or if there are extra turns, five additional turns are taken in total (assuming nobody wins before that).
So if the Andy casts Emrakul on the first extra turn targeting Nathan the turns will end up being like this:

Andy (casts Emrakul)
Nathan (controlled by Andy)
Nathan (controlled by himself)
Andy
Nathan

From the Magic Tournament Policy:

2.5. End-of-Match Procedure
If the match time limit is reached before a winner is determined, the player whose turn it is finishes his or her turn and five additional turns are played in total. This usually means that one player takes three turns and the other two, but a player taking extra turns may affect this. (...)