This question is addressed quite clearly in
The Stand, by
Stephen King. It thoroughly examines the society and environment formed by a
post-
apocalyptic society, in this case, one visited by a great
plague. There is no
holocaust: all the people are
dead. All our toys are fine.
But the question asked is, "How long would utilities last if everyone disappeared tomorrow morning?" Not, "What would society be like. Well, I'd have to hazard a guess at a few days, maybe. If everyone disappeared simultaneously, stoves would be left on, electric blankets left on, roasts left in the oven, jacuzzi's would overflow, spraying water everywhere, because the weight of the person holding down the water in it would be gone.
It would be havoc. Towns would burn. If everyone vanished, the questions stated above in RimRod's writeup would be entirely irrelevant. The question should be asked, "How long would it take our automation, which we foolishly believe would run on, to fail without our presence?"
But we'd be all gone, so it wouldn't matter anyway.