I have been looking for something similar, and my current solution is to use an RSS Client like Feeder, and to subscribe to the RSS Feeds of some quality outlets.

Results in a feed that looks just about like this:

Find out if they have RSS/Atom feeds and use any Feed reader

As was already mentioned, an RSS reader would be best. I like miniflux in conjunction with the News Android app.

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 😉

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.