I think the threat of an actual nuclear war is pretty small, and always has been. Even politicians were pretty quick to realize that nobody wins a nuclear exchange. Even if you were able to wipe out your enemy their territory would be useless for millions of years, and the damage to your own population, land and infrastructure would be catastrophic.
The real threat in modern society is the whacko with a backpack nuke who has nothing to lose, nothing to live for and is only interested in taking as many people with him as he can. Whether a religious fundamentalist (any flavor) racial bigot (also, any flavor) or simply a nutjob going postal, they can cause a significant amount of damage and are almost impossible to stop.
Of course, there are nut job national leaders (see North Korea or Iran) who could possibly launch nuclear warheads against their neighbors, but that would be a relatively small exchange and not likely to destroy much beyond their own general area.
Yes, disease can be a problem. But even the most virulent of plagues rarely has a death rate exceeding about 75%. Even if you postulate a 90% mortality rate, with a current world population of about 8 billion people the survivors would still number around 800 million! Hardly a wipe out! Of course, they would be spread out across the globe and civilization as we know it would probably be destroyed, or at least set back quite a ways. But with that many survivors, and with all of the accumulated knowledge of humanity still available, recovery would be steady, if not necessarily rapid.At any time, a devastating plague can reappear, just like the Black Death in the 1300's. More and more bacteria evolve multi resistance against antibiotics, what if the medical industry researching for new antibiotics turn out to fail? Sure, mankind survived without antibiotics for millenia, but an airborne disease can spread very fast in this day and age.
In fact, ANY kind of major catastrophe, other than the complete destruction of the planet or the destruction of the Sun, would invariably leave survivors, and in generally large numbers. Some of them, like nuclear holocaust or plague, would also have significant impacts on other species. But over all, the biggest threat is to our way of life, our civilization. And even then it's mostly a matter of a change in how we must live rather than complete destruction.