I noted that WC was reporting on Pocket Casts appearing as a 'hosted web app' for Desktop in the Microsoft Store, i.e. essentially an instance of Edge running a particular URL. And I thought it noting (below) that you can run essentially the identical system in Edge directly under Windows 10 Mobile. The interface isn't that much fun to use and you still have to pay $9 if you want to use the system after the initial 14 day trial, but if you run Pocket Casts on another device (e.g. an Android smartphone) then this is one way to have your podcasts and listening status fully synced and ready.

Of course, I'd still recommend that you use one of the many UWP podcatchers for Windows 10 Mobile instead. Even the new beta Podcast Lounge 2. But yes, if you don't mind coughing up a little to Shifty Jelly then you can use their servers to run a Pocket Casts 'instance' in a tab in your Edge browser:

The URL to copy/paste/type/hit is https://play.pocketcasts.com/web/podcasts/index#/podcasts/new_releases (or similar) and I strongly suggest using your phone in landscape mode, as here, since the display scaling for this web interface really doesn't work in portrait on a phone!

Your Pocket Casts account play status is fully shown and it's easy enough to use the left list to navigate and then just start listening.

Note that these solutions necessarily involve streaming everything, i.e. there's no download facility, though you can switch away from Edge (or even turn the display off) while listening, thankfully, so it's not as if the screen player has to be shown during playback. I did note the odd bug though and some podcasts had to be kicked off a couple of times before the audio actually started playing.

Talking of bugs, Shifty Jelly are still in my bad books for launching on Windows Phone 8.1 in a blaze of:

Welcome to the first update of many for our Windows app! Those of you who started countdown timers to see how long until we abandoned it are out of luck I'm afraid. It's nothing personal, but I'd avoid buying lottery tickets if I were you ;

... and then shipping one minor bug fix in three years. Very disappointing. I reviewed it here, back in 2015. Oh well.

At least the options are plentiful anyway. Pocket Casts in the browser, Pocket Casts in their old 8.1 application. Or use a modern UWP podcatcher. It really is up to you!