What if developing nuclear weapons was physically impossible?

I can see a few things right off the bat:

-Operation Downfall: The Allied invasion of Japan through conventional means. With no nuclear bombs to turn Hiroshima and Nagasaki into radioactive dust pits, the Allies would be forced to invade the Japanese mainland in early November. Mustering the single-largest Allied force ever created and the largest landing in warfare history, 6,000,000 American, British, French and Commonwealth troops would plow through an infernal nightmare unlike anything viewed by man. Over 35.8 million fanatic Japanese defending every inch of their country down to the last man, in suicidal attacks and displays of violence that would question each soldier's own humanity. The Soviets would later join the Allies by invading the north, possibly creating a Communist north and Capitalist south similar to what happened in Korea.

-Conventional Stand-off: The prospect of a world-ending conflict becomes far too impossible. The very threat of nuclear bombs exterminating all Earth-borne life within minutes no longer exists, meaning superpowers would rely on conventional warfare for staying at par with their opponents, rather than mass-manufacture of nukes. The US and USSR would engage in a fairly-comical Tom-and-Jerry-style battle for outmatching each other militarily, developing new warfare technologies and essentially booby-trapping the most possible invasion routes in case that an unwarned attack comes. The US initiates a massive upgrading of its military, setting course to replace its old WWII-era stack with a modern arsenal, including jets, rockets, massive carriers and better tanks. Both NATO and Warsaw Pact countries do the same. Neither the Soviets nor the Americans really wished to engage in conventional war, staving off conflict regardless of propagandistic efforts to paint the enemy in an evil light. 

Any thoughts on WWIII, though?

Wich result in countries deciding to keep much larger armies than they had previous, also more chemical and biologic weapons development.

As would the Soviets and anybody else i can assume, put them on rockets (if that is a handy deliverance system) and you do not need nukes.

Yes, a LOT more armed forces in many countries...
even if the OTL Cold-War-fronts exist (Nato versus WP), you need much more continental weapons and armed forces in the countries along the iron curtain as OTL.

OTL they knew, if "they" (meant are the "hostile other side") break trough we (the good ones) use nukes. 
Here you lack these ultimate weapons... so better safe then sorry.

Compared to OTL the WP will have maybe 50% more combat forces in the "border zone", the Nato will have 300-600% more forces. OTL they were quite thin, because they thought "we have time for the other Nato-Forces ("hello USA") to move them to europe". Here - that would be to late.

India and Pakistan would fight more often, China could or could  not face chemical weapons against its troops used by the US-armed forces in Korea. Maybe even against chinese cities... like Shanghai. 

Ironically the world would be not as safe as OTL, but only because nobody ever used the "button". Otherwise we couldn´t write here

Absolutly.... i wrote earlier that 10.000s of such monsterbombers exist, able to drop 50-100k tonnage of explosives to wipe out on the conventional way enemy cities.

But not only the big 5, also smaller states will either have them or will support them.
The Nato would still try to frighten the WP from attack, so the dutch, danes, germans, italians and others would have 1000-10.000 bombers each, either in their own arsenal or paying for that by some larger state (e.g. USA). 

With nukes things are much cheaper, if you don´t overdo it - like OTL.