I use the iPhone podcast app to get tons of podcasts, which I mostly listen to on my iPhone, but I have proved to myself I could then get them to my Apple Watch and that they played from my Apple Watch to my Bluetooth headphones paired to my Apple Watch.
HOWEVER, for the Apple Watch I prefer using a 3rd party podcast player, such as Overcast (but there are others), as it gives me better control over which podcasts are moved to the Apple Watch, and I sync them on demand, and do not need to do it while the Apple Watch is charging. iPhone "Watch" app -> Search -> "podcast" and you will find 3rd party podcast apps that work with the Apple Watch.