I assume someone must actually watch the programmes particularly when it's because of something recent in the news, as some references can be very obscure and might not even be in the script.
I mentioned Independent Television in Britain on the other thread but there's a bit in it about Lace, a mini-series starring Joan Collins, which ITV showed in the late eighties and they got loads of complaints about it featuring sex and violence at 8pm. LWT, who were responsible for playing it out, did an inquiry and found out that only two people in the company had actually watched it - the person who bought it, a year previously, and the head of compliance who said it needed editing but was on holiday when it was actually broadcast. Seemingly the schedulers saw the note about it needed editing but had assumed it had already been done. They tightened things up after that, though.
Sometimes stuff does get missed, though. The night in 1999 there was a bomb in a gay bar in Old Compton Street, Smack The Pony went out with a reference to blowing up someone's house. C4 did an apology after the episode saying it was "too late" to edit it out - which was surely not the case, because in those instances they would have dropped the show completely (and there was no mention of it before the episode, in a "this programme was recorded before the news..." kind of way). Seemingly they'd just completely forgotten about it. And BBC1 showed an episode of Only Fools and Horses that night with a scene in a gay bar, and had to apologise as well.
And on the day of the Omagh bomb, I remember BBC1 haphazardly snipping out the final sketch in a Harry Enfield repeat - it just cut from the previous sketch to the copyright board, missing out the end credits, so the news could start on time - but left in the "William Orangeman" sketch.