I was not able to download anything since I upgraded to iOS 8.4. What actually happened was that the podcasts would download but when the progress circle got full and displayed "Processing," I got error messages. After powering my device (iPhone 6) on and off for a total of maybe 5 times, which did nothing, I resorted to the following steps:
1. Back up my phone on iTunes.
2. Reset my phone (Settings > General > Reset > Erase All Content and Settings).
3. After the reset process and when my phone turned on, I chose to restore from an iTunes backup (the one I did in step 1).
4. After the restore process, the podcasts are back to normal. One more thing, podcast syncing, which never worked for me, is also working.
For more details on these steps, visit this: http://www.macworld.co.uk/how-to/iphone/how-reset-iphone-erase-content-reboot-re store-fix-frozen-3596892/