An update. It's still the case that the BBC podcasts can't be downloaded within Thunderbird but if I copy the link location and paste that into the address bar of a browser it works.

I have no issue installing BBC RSS feeds for news, but podcasts get me a message about "We're replacing BBC Sounds outside the UK and bringing you BBC.com, a seamless way to read, watch, and listen - all in one place."

I never get as far as finding an RSS feed to use. But I did read this https://www.reddit.com/r/podcasts/comments/1fdkno9/is_there_some_way_of_getting_rss_feeds_the_bbc/ so I am guessing that they are blocking you.

Hi Matt, It could be a location issue, when I hover over the attachment I get "is not found or is not accessible at this location anymore." But when I right click on that attachment I can copy the link location and paste into a browser address bar and it works just fine. So it looks like there's something about Thunderbird.

I'm in the UK so aren't getting the "We're replacing BBC Sounds" thing, more of a you WILL use it whether you like it or not! Auntie knows best and all that crap ;)

And the BBC RSS feeds are working as they should in Thunderbird. I'm assuming this was fixed by version 128.10.2esr.