Not sure if this would work for you but this app Podcini is new on F-Droid, and BBC is mentioned.
Podcini is an open-source podcast manager and player that allows you to subscribe
to any RSS feed, gives you instant access to millions of free and paid podcasts,
from independent podcasters to large publishing houses such as the BBC, NPR and CNN.
Add, import and export their feeds hassle-free using the Apple Podcasts database, OPML
files or simple RSS URLs.
Download, stream or queue episodes and enjoy them the way you like with adjustable
playback speeds, chapter support and a sleep timer.
Not sure if this would work for you but this app Podcini is new on F-Droid, and BBC is mentioned.
Well, it’s a podcast player. The BBC has podcasts. Don’t think that was what op meant.