Rather than wait for the BBC, Danny Baker has started his own podcast (it's also available through iTunes). And, unlike the current BBC podcasts, it does includes some music, albeit a rather eccentric selection.
So far he's only done one of them , but says that he hopes to do it on a daily basis before too long, and seems to imply that he may give up his daily show on BBC (Radio) London. He started his radio career there back in the late 80s (when BBC Radio London was called GLR), and always seems to to return there after his spells elsewhere (Radio 5, Radio 1, Virgin, TalkSport - see Wikipedia).
If I were cynical I might say that he has built his career on being able to take the same material and recycle it endlessly on different stations with slightly different audiences. So doing a podcast sounds like an inevitable career move. Or maybe it's just a way to prompt the BBC into doing a podcast of his existing show.
Danny Baker is probably something of an acquired taste - he is loud and brash, and indeed the first time I heard him on the radio I quickly switched over to something else - but I did eventually change my mind. If you can get through the first 5 minutes then you should be OK - he is a natural broadcaster with a phenomenal knowledge of trivia, so it's worth persevering.
For the first podcast, his "sidekick" is David Kuo, who apparently lived in Hong Kong during his childhood - and so one of his anecdotes revolved around his family's maid bringing home a live chicken from the wet market. Needless to say, Baker found almost every element of that tale rather bizarre. Well, I suppose he would. I'd rather have fewer sidekicks and more Danny Baker, but maybe that's just me.