We know the basilisk's stare must be viewed directly to retain the deadly effect, so at best it would petrify people watching TV, **but what of sounds?
Can a magical deadly scream such as the Mandrake scream,the Banshee's scream, or the rooster song in the event of a basilisk, retain their effect if they were recorded and then broadcast on the wizarding wireless network?
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.
Magic basically works via intent of the caster and direct transmission of magical energy (space and time matter in magic). Spells are visible in the air between the target and the wand for a reason.
I wouldn't even compare the radio to cameras and mirrors. In the case of reflecting or refracting surfaces, there is still (obstructed) line-of-sight for the magic to work over. It's still light, it still works off the same physical principle. But TV or radio... that would be a different story. The information gets converted by equipment to a completely different carrier. I'd say that TV/radio transmission protocol can't carry magic, only the harmless (maybe annoying) physical component of the scream/stare. Even in the muggle world, you can't transmit UV over TV, or ultrasound over radio (unless the protocol was designed to do so specifically). So we have to assume that the magic (the spell that's carried by the sound or the stare) is not converted into radio waves and back.
Even over a magical carrier signal (wizarding wireless network) you would also have a hard time transmitting this. Essentially, it would be have to recast the spell on the receiving end, which should then work to transmit other kinds of spells. For radio, that is against basic conservation laws - even with magic, it should take more effort to cast a spell on more people. The number of receivers in radio transmittion is basically limitless, so this would multiply the magical energy without an apparent source. The same (worse) problem is the possibility of recording and playing an infinite number of times. That shouldn't work at all. It's sort of like showing you a photo of a basilisk (if one could take a photo of course). You probably wouldn't get petrified. However, this also offers a clue to how this works: the spell cannot be transmitted to film, but destroys the equipment instead (Colin's camera). So translating this to the mandrake, its power might explode the microphone and fry the recording equipment, but wouldn't pass through.
I'd say we are safe.
