If I play Baron Nashor to Baron Pit with The Exile, does Baron come in ready?

Yes — as long as he was played to a battlefield you control, he enters ready. The Exile's "units you play to battlefields you control enter ready" is a replacement effect on how the unit enters, and it keys off where he's played to, not where he ends up.

Baron's "If you do, I enter there" is a separate replacement effect that only rewrites the entry location to the Baron Pit (CR 369.3, and FAQ #12610 makes the same play-location-vs-entry-location distinction for Here to Help). Both replacement effects apply to the same enter-the-board event; since you control Baron, you choose the order (CR 372), and the result is ready either way. The Pit being uncontrolled at the instant he lands doesn't matter here — that only affects Contested/Conquer, and you'll still Conquer the Pit after the Non-Combat Showdown.

The one way to miss the bonus: he wasn't played to a battlefield you control — e.g. you played him to your Base, or an effect fixed the play location (Rift Herald's "play to your base"). Then The Exile's condition isn't met and he enters exhausted.