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.

Annoyingly while #BBC has a #mastodon presence https://www.bbc.co.uk/rd/blog/2023-07-mastodon-distributed-decentralised-fediverse-activitypub it doesnt (yet) cover #bbcnews which i would have thought would have been a good starting point. There are already so many bots providing summaries and links to.#bbcnews.

I recently tried #rssparrot to add other #rss feeds from other news sites with some success.

Thanks for the intro to Feeder. It's changed how I read news. No more tabs and bookmarks 😉