I’ve given up. The Apple Podcast app, part of iTunes family, promises to sync across other devices (iPhone, iPad, etc.) but not only does it no sync, but in the recent week after a system update (pushed out by Apple), my subscriptions were lost. Lots of messing around with Google didn’t get anywhere.
So, I have deleted this app. Gone. No more.
Now using Downcast, available on the Appstore. Appears to work just fine.