There are two answers:
- For iTunes and other directories—these are usually cached and when everything is working properly, it can take 15 minutes to a few hours for a new episode to show up in the directory listing.
- For subscribers—with a few exceptions (Stitcher, Google Play, iHeartRadio, and Spotify), when someone subscribers to your podcast in an app, they are subscribed directly to your RSS feed. Thus, whenever you publish a new episode, it is immediately available to your subscribers. Some subscribers will have the apps refresh feeds immediately or at certain intervals. As long as the new episode is in the feed, subscribers can download it the next time their app refreshes.
You're not using FeedBurner or any special caching system, so I'll say this for any other readers. If your feed it cached (especially if you're using FeedBurner—even in its vanilla state), this introduces an additional delay.
Lastly, invalid podcast feeds (with incorrect data, illegal characters, or other problems) will prevent most podcast programs from updating the feed. Here's how to validate your podcast feeds.