Q: iPhone podcasts not working

This often happens when the podcast episodes are in the process of being updated/uploaded by the producers, but try a simple Reset. You can also try resetting your router.

- Reset: hold down the Home and Power buttons until you see the Apple logo (10-15 seconds).

Are you able to download any other files from iTunes Store?

The next step after Reset is to Restore.

- Restore your iDevice: https://support.apple.com/en-us/HT204184

If your backup is to iTunes, make sure it's encrypted.