This app was mentioned in 4 comments, with an average of 1.00 upvote
For those who want to listen on a computer:
www.bbc.co.uk/programmes/b006qfvv
If you want to listen to it on your phone (I only know about android), you need to get the BBC Media Player:
https://play.google.com/store/apps/details?id=air.uk.co.bbc.android.mediaplayer&hl=en
and then go to the first link to listen on the mobile page.
If you mean for mobile, there is a solution. On Android there is a separate app called BBC Media Player that allows users to play the BBC sound media. Also on iOS there is some internal jiggery pokery that goes on to allow otherwise Flash-based BBC recordings and streams to play.
It's actually a well designed app too. I wish it had a sleep timer though. You can also install BBC media player from the play store (not geo locked anymore):
https://play.google.com/store/apps/details?id=air.uk.co.bbc.android.mediaplayer&hl=en
And then visit the website (http://bbc.co.uk/radio) and it will play the programs directly.
It works fine for me also. Do you have the BBC Media Player app installed on your phone? Perhaps that is required.
https://play.google.com/store/apps/details?id=air.uk.co.bbc.android.mediaplayer&hl=en