There's currently no way to access the Queue while listening to a podcast.
However, you can still add podcasts to the Queue. That's pretty weird.
You can line podcasts up in the Queue to listen to, but you can't actually view the Queue- and there's no ability to skip a podcast that you've queued up.
You're just firing them into the Spotify ether, to creep up on you on some random Tuesday morning while you're in the middle of a traffic jam, and had forgotten that you queued up an episode the day before that you *really* don't feel like listening to right now.
Just change the 'Share' menu option in the bottom right screen 'Go to Queue'.
Exactly like it is when you're listening to songs.
Or, if you really wanted to keep that 'Share' option (honestly, who shares podcasts???), make that bottom righthand corner into a 'Menu', and include both 'Share' and 'Go to Queue' in the options.
It's not hard, guys.
Don't be weird.
Updated on 2019-08-10
Hello and thanks for bringing your feedback to the Idea Exchange.
A similar idea has also been suggested here: https://community.spotify.com/t5/Live-Ideas/Podcasts-View-the-Play-Queue-While-Listening-to-a-Podcas...
We have also merged the votes from this idea to that original one. More information about how the Spotify Idea Exchange works can be found: here