I had a call with Apple support, who may have helped me solve this issue.
They suggested turning off WiFi on my phone, and then trying to sync. I did this, and as a side effect, my watch then prompted me to enter my Apple ID password. I had to type in my password on the little keyboard on the watch.
Since I did this, music and podcasts are synchronising just fine. I had to clear down the albums/playlists that were previously stuck, but I've just checked and my podcasts also synchronised overnight while the watch was on charge. Note: Music seems to sync when the watch is on battery, but podcasts require it to be charging.
So I would suggest trying the above, or another way to force iCloud to 're-authenticate' might be to do something iCloud related on the watch by going to Settings > Your Name > Password & Security and try changing your password, or manage a subscription or something (just a guess).
Apple support also had me sign out of iCloud and back in again on my phone, which didn't seem help. Mentioning it here in case it did in some way prompt that password prompt on the watch, but it didn't seem to be related (it only started working a few hours later when I disabled WiFi on my phone and tried re-syncing).
Hope this helps, let me know if you have any success.