I had the same problem, except I don't have the Apple Watch. I'm running an iPhone 8+, IOS 13.2.3. When I set the settings listed below, it began to work again. I don't think that I was able to fix the problem, just seemed to be a good work around. Please note, that I sync my podcasts from iTunes only. I do not do any downloads directly to my iPhone. Apple Support was not able to help me at all. Once I disabled the Cellular Data and the problem was fixed, they immediately took credit for fixing the problem.
Background App Refresh -> off
Cellular Data -> off
Sync Podcasts -> On
Block Downloads Over Cellular - off
Continuous Playback -> On
Refresh Every -> Manually
Download Episodes -> off
Delete Played Episodes -> On