I hope that you won't mind me going off at a tangent but I'll point our that I listen to Round the Horne, Beyond Our Ken, The Navy Lark, etc every day of the year for free. Rather than downloading them I simply listen to them being streamed on the various ROKiT Radio channels:
https://rokitradio.com/
Their Old Time Gold channel only carries British comedy content between 0000 and 0300 GMT and from 1200 to 1500 GMT. (Remember to add on an hour for BST). Content on that channel includes Hancock's Half Hour, I'm Sorry I Haven't a Clue, Take It From Here and Dad's Army.
The British Comedy One channel broadcasts 24 hours per day but, following some server problems a while back, they're currently not following their published schedule. (So it's pot luck as to what's on at any particular time). However you'll find programmes like Beyond Our Ken there.
The British Comedy Two Channel also broadcasts 24 hours per day (and does follow the published timetable), with content including The Men From The Ministry, The Navy Lark, Just A Minute, Hinge and Bracket, The Clitheroe Kid, The News Quiz, The Goon Show, etc.
Abacus Radio might also be of interest to you:
http://www.abacusradio.com/
Their British Comedy Channel doesn't have a published timetable but it's on 24 hours per day and it's home to The Burkiss Way, Yes Minister, Hancock's Half Hour, etc.
The Abacus Goon Show Channel, unsurprisingly, broadcast episodes of The Goon Show 24 hours per day.
Another source of 24-hour Goon Shows is GoonShowRadio:
http://goons.fabcat.org/
(Click the 64Kbps link and the content will open in your default media player, rather than directly in your browser)
With all of the above services, just like with a traditional radio station, you have to accept what's currently on offer when you want to listen. (i.e. there's no 'on demand' facility). RadioEchoes is different though, in that it's a repository of old radio programmes that you can either stream on demand or download to keep and listen to whenever you want:
http://www.radioechoes.com/
Most of the content on that website is American but, if you hunt around a bit, there are some British shows too, For example, you can choose between 153 different episodes of The Men From The Ministry
http://www.radioechoes.com/?page=series&genre=OTR-Comedy&series=The%20Men%20From%20The%20Ministry
and there are no fewer than 247 episodes of The Navy Lark available there
http://www.radioechoes.com/?page=series&genre=OTR-Comedy&series=The%20Navy%20Lark
Hopefully that list will give you a few things to listen to ;-)