Regular BBC pages to check :

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BBC Category : Folk
Here's the BBC page which gives links to the current Weekly, Monthly & Seasonal radio programmes.
(There are also a few one-offs on the page.)

Ambell i Gân (Cymru)
A' Mire ri Mòir (Nan Gàidheal)
* Blas Ceoil - (Radio Ulster)
Caithream Ciùil (Nan Gàidhea)
Catgut and Ivory (Scotland)
Celtic Heartbeat (Wales)
Claire and Friends (Scotland\Shetland)
Crùnluath (Nan Gàidheal)
Fae Hameaboot (Scotland)
Folk Club (Ulster)
Georgia Ruth (Cymru)
* Iain Anderson (r2)
Music Planet (r3)
Oota Da Cans (winter only) (Scotland)
Orkney Folk Music with Jennifer Wrigley (Scotland\Orkney)
Orkney Irish Country \What's the Craic with Colin Kirkness (Scotland\Orkney)
Orkney Scottish Dance Music with Liam Muir\Dashing White Farmer (Scotland\Orkney)
Pipeline \Pipes and Drums (Scotland)
The Piping Season (Scotland)
Piping Sounds (Scotland)
Roddy Hart (Scotland)
Saltfish Sessions (Scotland\Orkney)
Take the Floor - Gary Innes (Scotland)
The Folk Show With Mark Radcliffe (r2)
Tim Walker's Folk (Lincolnshire)
Tiompan (Nan Gàidheal)
Trad Ar Fad! (Ulster)
Travelling Folk (Scotland)

- NB This link might try to open your 'Sounds' app which might not give the same information as the web page - you might need to copy it and paste into a browser
www.bbc.co.uk/sounds/category/music-folk?sort=title
[not blickified, on purpose]

*regular programmes missing from the page above :
Iain Anderson (country, folk, blues and soul) - Radio2
www.bbc.co.uk/programmes/b0074hkv/episodes

Blas Ceoil - Radio Ulster
,,,new releases, gig guide details and live acoustic sessions to keep. (in Gaelic; quite a lot of chat)
www.bbc.co.uk/programmes/b00cmsk6/episodes
& check the 'clips' tab for more.

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Available For Over A Year
There is a separate mudcat thread listing BBC radio programmes 'available for over a year'.
mudcat.org/thread.cfm?threadid=171922

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search : Folk
Also of interest might be this search page which gives 20+ pages of radio and TV.
But it isn't sortable, doesn't indicate which pages have available programmes.   ,,, & includes news items e.g. about Folkstone!
It's useful to see what the BBC has broadcast in the past though.
www.bbc.co.uk/search?q=Folk

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Category : World - radio
Current shows; it's a wide interpretation of 'world' plus a few extra programmes.
- open in a browser -
www.bbc.co.uk/sounds/category/music-world?sort=title   

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Genre : music : World - radio & TV
www.bbc.co.uk/programmes/genres/music/world/player
There are filters for
Africa
Americas
Asia Pacific
Cross Cultural
Europe
Experimental & New
Middle East & Africa