If you've not downloaded anything recently it can take 5 minutes or so for the first item to start downloading, once started subsequent downloads will start immediately.
If you are still having issues then there are a few things to check:
- You haven't accidentally paused the download - tapping on the download icon will pause/resume the download.
- You have enough space on your phone or iPad to save the show.
- You aren't low on battery or in low power mode - plugging in to charge when you're low on battery will help.
- That you are on a fast and stable WiFi connection - poor data connection will cause your download to pause.