It does take up to 30 minutes to get the latest live show ready to be made available as a download. If it has been more than 30 minutes since the live show ended refresh the show in your podcast app, usually by opening the show in the app and drawing your finger down on the screen.
Also, check to make sure you are still subscribed to the show. If you are not subscribed within the app, it will not download new episodes. If you haven’t listened to any downloads recently, the app will stop downloading shows automatically. You can always see whatever shows are available for download by clicking AVAILABLE EPISODES to see a list of recent shows. Just click on those show titles and it will download the shows for you.