Making nuclear weapons is much easier than it used be. Germany and Japan could both enrich Uranium from their civilian nuclear programs in weeks or months if they wanted to. (but nobody is worried they will, largely because of their allies and the history of them not doing so)
(we 'stop' other nations from acquiring weapons by offering them access to civilian nuclear power - via the NNP treaty - but this didn't work to stop India or Pakistan.)
That said there are two difficult steps - enriching Uranium, with centrifuges operating on Uranium gases (probably Uranium Fluoride) and once you have enriched enough a bomb is really easy to make (rockets are still much more difficult, but i'm sure another delivery system could be used)
Plutonium weapons also involve a difficult step, it is much easier to get your hands on Plutonium (from a nuclear reactor) but requires an implosion which is very complicated (though i guess modern computers could simulate the required shock-wave fronts... )
The solution? Support states, have them included in the international community to the point where they have more to lose by being left out.
The Marshal plan is a great example, the US supporting the rebuilding of Europe (because what happened after world war 1 in germany was a terrible result of that war) and including western europe (along with Japan, as it happens) in the American 'empire' a commercial enterprise - This allowed Europeans buy US goods and defend themselves from the Soviet Union (i'm sure different parts of the American political system were more or less interested in each of these goals, but both goals were achieved)
To include nations like Iran and North Korea (well you might want first to resolve the Korean war, and unify North and South... ) in the international community, give them more so they don't have more to lose from war than they can possibly gain...
China is not a threat to 'world stability' because it is thriving on the trade relationships it has with the US, Russia, Africa, and Europe.... How to include Korea or Iran is a difficult question, but that is what you should be asking.