How long does a batter have when the umpire says Batter up?

Answer 1

The MLB Rule says:

6.02(a): The batter shall take his position in the batter's box promptly when it is his time at bat.

6.02(c): If the batter refuses to take his position in the batter's box during his time at bat, the umpire shall order the pitcher to pitch, and shall call "Strike" on each such pitch.

So there's no rule that says how long the batter has to step into the batter's box, only that he do it "promptly." But as an umpire, I can tell you the umpire will not put up with much messing around on the batter's part. If he tells the batter to step in, and he doesn't, the batter will get penalized.

Answer 2

He is supposed to make his way from the On-Deck Circle to the batter's box immediately after the last play has ended. The time limit is based upon the Umpire's sole decision. He will motion the batter to the box if he doesn't go in a timely manner and after that, if the batter doesn't make his way to the box, he will call "Strike." He can do this until the batter is struck out.

So there's a good trivia question for you. How can a batter be struck out without having a single ball pitched to him? That's how.