Why isn't auto download working for the iphone Podcast app?
Go to Settings, then scroll down and tap on Podcasts app icon:



So I'm assuming you turned "Auto-Downloads" on...

Then tap "Episodes to Keep" and specify which episodes you want to keep. (Depending on your listening habits, I'd not keep this setting on "All" because it can quickly eat up your available memory. Mine is set on "Last 3 Episodes" which gives me plenty of time to listen to them.)

Then go back. To address your issue you may want to turn on "Use Cellular Data" which is Off by default. What might happen in your case is that your phone disconnects from WiFi when it's not on. (This is done to conserve battery.) But in this case this setting will prevent the Podcast app from auto-downloading. Keep in mind though, that turning this setting on can quickly use up your cellular plan's data limit. So check with your cellular provider if you have enough data quota to last your billing cycle without incurring additional charges.