While I am generally against doomsaying, there is a reasonable argument in favor of the position that a nuclear war will likely happen in the foreseeable future. Whatever can happen sometimes does, in the sense that something that is not decreasing in probability over time will be sure to happen sooner or later. I do not see good reason to suppose that the probability of nuclear war over a fixed period of a year is going down in any significant way; and it may be increasing, due to the availability of the technology to an increasing number of nations, the increase in wealth leading to a greater ability to produce them, and so on.
The question is whether that probability is so low that our reasonable expectation would be that it pretty definitely would not happen during our lifetimes, like the probability of the earth being hit by an asteroid such as the one that destroyed the dinosaurs. The latter probability is so low that there would be no point in worrying about it.
However, nuclear war would be a human and political event, not a natural disaster, and should be judged differently. Since nuclear weapons have already been used, implicit and explicit threats of the use of nuclear weapons have been made time and time again, and there have been mistakes that could have started a nuclear war, it would be unreasonable to assign an extremely low probability to such an event. I would suggest that a chance of 1% per year would be a reasonable guess. If so, there is about a 40% chance that there will be a nuclear war within the next 50 years.
If this is a reasonable assessment of the probability, it is not something to simply dismiss. Pope Francis, citing Pope Benedict XVI and Pope John XXIII, argues for some kind of world government:
As Benedict XVI has affirmed in continuity with the social teaching of the Church: “To manage the global economy; to revive economies hit by the crisis; to avoid any deterioration of the present crisis and the greater imbalances that would result; to bring about integral and timely disarmament, food security and peace; to guarantee the protection of the environment and to regulate migration: for all this, there is urgent need of a true world political authority, as my predecessor Blessed John XXIII indicated some years ago”
“Integral and timely disarmament” likely refers at least in large part to nuclear weapons. The argument would be that the only way to reduce the probability of nuclear war to a reasonably low level would be to completely eliminate such weapons; and there is no reasonable possibility of doing this without a world government. For even if all current nuclear states eliminated their weapons, there would be a greatly increased incentive for small countries to build them, since it would provide them with a definitive advantage over their neighbors which currently does not exist.
In principle there could be other ways to resolve this problem, but there is no guarantee that such a solution will be worked out quickly enough in practice.