Let's start off on the right note, and thank you for those to whom this doesn't apply: the past and passive form of "to hang" as in "death by hanging" is "hanged," not "hung." In addition, hanging is quick, if it works as expected. Ideally, the noose snaps the victim's neck immediately. Every once in a while there is a case of somebody with a thick neck ending up strangled by the rope instead. As for the other things mentioned here, these are the facts:
The United States is not executing Saddam Hussein, the High Tribunal of Iraq is, and even though it is supported by the United States, the court itself was set up and is backed by the elected representatives of the Iraqi people. If hanging sounds outmoded, remember that this is an outmoded society. This goes back to the fact that Saddam was tried under Iraqi law, which is outmoded, not United States law.
Excepting a legal miracle, Saddam will be hanged in the coming weeks. Under Iraqi law, he has 10 days to appeal. The appellate court is expected to uphold the ruling, which means, under law, he must be executed 30 days after that. If the appellate court overturns either the conviction or sentence for crimes against humanity, remember that he is undergoing another trial right now for genocide, and will unlikely escape that as well. If he does, there is a whole slew of charges that can possibly be brought against him, most of them being capital crimes.
As for televising it, I wouldn't be surprised if a cable network did that, because cable networks are not governed by the FCC, but I doubt it would be live; however, I don't think that if it is televised, it won't be limited to one station (I've heard rumblings of Fox), but rather almost all the cable stations will because it is expected to have a rather large audience, if in fact it is on television, and they want their ratings.