I can use the watch as a remote for playing the podcast on the phone which is great but I would like to just be able to use the watch but this doesn't seem to be possible.
If you have the BT version of the Galaxy Watch you are depending on having your Phone nearby to listen to PodCasts streams anyway, so I'd say problem solved (listen to the podcast through your Phone)?
For the LTE version of the watch I don't think there is is an option to listen to podcasts directly. The Spotify app is often mentioned, but the Spotify app for the Galaxy Watch doesn't have that option yet as far as I know.
For both versions: what I would try is to find a podcast app on your Phone that allows downloading the podcasts as audio file, transfer the audio file to your watch and then listen to it from your watch.
Haven't tried it, but that may be an option?