I will try an educated guess, since not that much informations are given, whether on Mandrakes or on the Wizarding radio.
When Mandrakes are young, it is said in CoS (Chapter 6) that their shout can knock people out. We can imagine that it is because their lethal capacities come when growing up (I think it's in Chapter 15 or 16, when Ms Sprout explains that they will be soon able to use Mandrakes once they lose their acne, meaning they seem to age in a matter that can be compared with human one). As you said, the Basilisk lethal gaze can be somehow reduced to a "simplier" paralyzing gaze when you look at him in other ways than the naked eye.
Concerning the Radio, I think we can agree that listening to the radio isn't quite the same than the naked ear. You have the micro, the whole stuff, the magical wave to broadcast sound, then again the radio set, the speaker, and then only the listener's ear.
So, considering the example of the Basilisk and the fact that the Radio can be compared with mirrors, ghosts or camera, I think it is reasonable to say that a Mandrake's shout recorded and broadcast on the Wizarding Radio won't kill the listener, but are more to knock them all out, which could be in my opinion a great April fool from broadcasters... Or whatever use you like.
Once again, these are but assumptions made on what we know, and I don't think JKR ever wrote a full article on the WWR and on how does it work.