This Radio 4 programme about Cockney changing and dying out is really good for language change & variation.
Lexicon Valley is a regular linguistics podcast. Have a look here for a list of recent episodes.
Lingthusiasm is a new linguistics podcast and you can find out more here.
Michael Rosen's Word of Mouth on Radio 4 has loads of good programmes about language and a few are listed here:
American English: http://www.bbc.co.uk/programmes/b08g5533
Attitudes to accents: http://www.bbc.co.uk/programmes/b01s4g7g
Emoji and technology: http://www.bbc.co.uk/programmes/b08ffvp6
Language pedantry and discourses: http://www.bbc.co.uk/programmes/b05qgch7
Child language & interaction: http://www.bbc.co.uk/programmes/b05077ks
Children's first words: http://www.bbc.co.uk/programmes/b04v382j
Office jargon & occupational English: http://www.bbc.co.uk/programmes/b03phrwl
Baby Talk and language development: http://www.bbc.co.uk/programmes/b03mfwjz