To view the Up-Next queue: you need to use the "full-screen" player on the iOS device. If you've just got the little player at the bottom of the screen - the single narrow bar, right above the Unplayed / My Podcasts / etc. buttons - then you need to press the middle of that bar. That should expand to the full-screen player.
Once you've got the full-screen player, look in the lower right corner, to the right of the "Jump 15 seconds" button. You should see a button with three right-bulleted lines. Press that button. That'll show you the Up Next queue.
Making changes in Up Next? That's a lot harder, absurdly so in my opinion. I don't think you can change things directly from that list. You can add other podcast episodes to the Up Next list, either with "Play Next" (which puts it at the top of the queue) or "Add to Up Next" (which adds it to the bottom of the queue). That'll work if you know where your episodes are, and you're willing to do this one episode at a time.
Or: make a station with manual play order. Go to "My Podcasts", press "+", Add a Station. In the Station Settings, set "Play" to "Manual". Then, when you go into that station, you can drag and drop the different episodes to the order you want. This is the best way I've found to construct a playlist in manual order, but man, they don't make it easy.
I keep hoping there's an easier way that I'm missing. The way this is currently set up is absurd.