There's an interesting question of why they're needed. Rationally, global thermonuclear war is a non-starter. If war is politics by other means, then there would never be a large scale strategic nuclear exchange because the consequences of doing so would zero out any possible gains, unless a nuclear power gained some capability to make sure that a first strike could not be retaliated against (to go sci-fi, imagine something like Goldeneye: a weapon that could cause a technological blackout on your target before you launch your weapons, meaning they wouldn't see it in time to launch a counterstrike).
Limited nuclear exchange is likewise loaded with more costs than benefits. Their use in World War II was a one-time thing because it was so shocking and unprecedented that it scared the Japanese into surrender, and was an example of strategic and not tactical use. Tactical use would presume a doomsday scenario (like Israel's nuclear plan, which is designed to spite anyone who overwhelmed Israel through conventional military strength by salting the earth of the Promised Land), where you're risking immense environmental damage in order to secure a battlefield victory, such that the irradiated area would not be safe for use in the war or immediate postwar period. If you're fighting a war over land, it's probably a good idea to not nuke the land you're fighting for. Even if you're not, the sheer destruction would play hell on your supply chains and logistics.
There's good research out there on the idea that the threat of nuclear war is a deterrent to conventional war, but in this day and age conventional war is its own deterrent; it's costly and messy and often politically unpopular; it's not the tool it once was for politicians to distract people.
The other question of total disarmament is: how would we be sure everyone kept faith? Well, we couldn't be, but it's not like the knowledge of building nukes would be gone from the world. If someone rearmed, the world would know and be able to react accordingly.