Everyone declares war on everyone else.
All jokes aside, ultimately clean nukes probably won't change the equation much.
Without the possibility of turning the world into a radioactive wasteland, it would seem like the use of nuclear weapons would become more politically acceptable, but...
Thermonuclear weapons are actually extremely clean (relatively speaking) and won't generate radioactive fallout if the fireball doesn't touch the ground. The Tsar Bomba test was actually one of the cleanest nuclear weapons ever detonated (that of course would change if it was detonated at its full 100 MT design yield).
Purely clean fusion bombs that don't use a fission primary are a theoretical possibility and have been researched heavily, but there are still engineering challenges that remain. Most theories involve muon catalysts or explosive pumped magnetic fields to compress the fusion fuel.
If the USA, Russia, or China did succeed in creating such a clean weapon, it's unlikely they would be public about it to avoid triggering an arms race or create tensions that could lead to a preemptive attack (after all, radioactive fallout is the main political problem surrounding nuclear weapons, even if it is a red herring).
But at the end of the day, it's unlikely that even if all sides knew of and had these clean nukes that things would change for one simple reason: Mutually Assured Destruction.
MAD actually works quite well when state actors are rational, as is clearly shown from the Cold War's lack of an actual nuclear conflict. Clean nuclear weapons are still devastating nuclear weapons, and a full on assault between two nations would still leave them in tatters within minutes, so the deterrent effect, which is what nukes are really designed for, remains, even if the nukes are clean.
However, what is a more dangerous proposition is if all nuclear weapons were magically removed from the planet. This is the true "Everyone declares war on everyone else" situation, as it returns us to a situation where wars between great powers become winnable.
The USA, Russia, and China would be far more likely to get into a conventional war knowing there was no possibility for it to go nuclear. Similarly, smaller states without nuclear weapons might be more tempted to take on larger states with them, knowing that the possibility of mass casualties from nuclear attack no longer exist.
Ultimately, it is the destructive capability over time states can bring on each other that acts as a deterrent. This is very evident in the situation on the Korean peninsula. Even without nuclear weapons, the fact that Pyongyang can absolutely devastate Seoul with artillery attacks has been a major deterrent for action from the South and the USA for decades. The fact that North Korea has been able to produce nuclear weapons is more for allowing it to force concessions out of world powers (after all, you can ignore artillery on a border that is ineffective beyond 30 miles or so, but not the possession of deliverable nukes that can destroy cities on a global scale).
There is also the prestige of being a nuclear power, as evidenced by the joke: "What do you call a country with nuclear weapons?" "Sir."