Copies of the weekly magazine the "Radio Times" from the 1940's and other decades can be read free online. BBC Genome " This site contains the BBC listings information which the BBC printed in Radio Times between 1923 and 2009. You can search the site for BBC programmes, people, dates and Radio Times editions. We hope it helps you find information about that long forgotten BBC programme, research a particular person or browse your own involvement with the BBC."
I looked through this the other day and found plenty of interesting items. Obviously the listings are not so heavy on detail, but they did throw up some possible research avenues to explore. An example would be a discussion programme back in August 1946, where an array of speakers discussed their views and relationship with Orde Wingate.
Glad you like it, I've just found it today. You can submit an edited version of the computer-generated modern transcription for approval but you can't see the original text as you edit it. If you find this frustrating I can recommend hours of editing fun with the digital newspaper and more website at Trove. On this site you see the original text highlighted as you correct the transcription. And you get points and a ranking. Since Australian newspapers printed news from the whole world through the usual press syndicates such as Reuters and Associated Press this is a marvellous resource and very accessible.