I'm not really sold on the idea that nuclear weapons are a salve against conflict. The only real examples are North Korea not getting invaded and maybe tamping down some of the India-Pakistan conflicts (note that India and Pakistan have had armed conflict with each other since both acquired nuclear weapons!). On the contrary, we can see times where nuclear weapons seem to have emboldened leaders--Putin has definitely played the "you can't retaliate against my aggression, I'll nuke you" card, and the Cuban missile crisis had both the American and Soviet militaries stridently arguing for a nuclear first strike because going second would be so calamitous (it was avoided largely because both Kennedy and Khrushchev were reticent to do so).

One of the striking features of the last decade or so of global politics is the willingness of middle powers to arm and fuel regional conflicts, so I think it entirely plausible that a world where middle powers like Saudi Arabia and Iran gain nuclear weapons is one where, say, the Sudanese Civil War involves nukes.