If I put a hammer over your head that can fall any minute you'll be worrying, but if you're born with the hammer over your head and your parents before you as well, it becomes less of a thing.
My parents had no problem reminding us that we all live with a nuclear sword hanging over our heads.
It just so happens that most people in the West are comfortable, are completely insulated from the consequences of war, and can't even imagine a regular war happening to them.
And nuclear war is so much more horrifying and its consequences are so much beyond the pale, that people can't even think of what it would mean.
No, it’s simply the end of the cold war that made it a possibility less present in the media. The cold war was cold because making it hot would have meant going nuclear. So the possibility was always closely linked to the state of cold war. Globalization has blurred the picture considerably.
maybe your parents aren't old enough to remember how much of the population could expect to die in wars before nuclear weapons (i.e. mutually assured destruction) existed
On an individual level, we all have a variety of hammers over our heads. Cancer has killed far more people prematurely than nuclear weapons. Something like 500,000 people a year are murdered. Traffic/bicycle/pedestrian accidents also kill more than nuclear weapons. Even compared to a once-in-a-century nuclear war that perhaps kills a billion people, cancer will kill roughly a billion in the next century anyway. So, for the rational/selfish person, the nuclear threat isn't worth worrying about.