The US doctrine on nuclear stuff is (paraphrasing here) "if you want something done right, you have to do it yourself." There are very few countries that we trust enough to hand over a job as important as nuclear deterrent. Australia would be a much better choice from an allied perspective but their military is much too small to be a real deterrent against China. Combining Australia, Japan, and South Korea might be an effective deterrent, but we still don't trust Japan and China will never let a nuclear South Korea happen.

Letting Japan have nuclear weapons is many decades in the future. The US monitors/controls the development of the JSDF very closely. They were only recently [1] allowed to have an "aircraft carrier." (edit 2: This presupposes that Japan even wants nuclear weapons. They, of all countries, would think twice about something that has destroyed two of their cities- also the "peaceful" application of nuclear power has devastated the area around Fukushima. New Zealand has a strong anti-nuke sentiment and they were only used as a test site.)

South Korea is so close to China that their defensive reaction time is effectively zero. The same goes for China's reaction time from a South Korean missile. China would be extremely upset (to the point of sabotaging missile installations) if we were to allow South Korea to develop nuclear weapons. Imagine if China allied with Mexico and started helping them with a nuclear program. The US would go apeshit. (edit: Rather than make up a hypothetical, just look at the Cuban Missile Crisis to figure out how crazy it could be. Why did I not just use that example in the first place?)

The US wants to control its own nuclear weapons and would rather lease bases from allies than to let the allies have their own weapons. This increases the perceived security of the weapons (from being stolen), as most military commanders believe that their own base is a much more secure place to keep nukes than someone else's base. Whether or not this is true is up for debate. Manning your own defenses instead of an ally also increases your perceived defensive readiness. Most commanders also believe that their own troops are better prepared than someone else's troops. Whether or not this is true is also up for debate.

[1] https://en.wikipedia.org/wiki/Hy%C5%ABga-class_helicopter_de...

A nuclear Japan would create instability and could easily lead to war in the region - possibly with Japan as the aggressor; at least that is what their neighbors see. There is still great hatred and distrust of Japan in SK and China, in part because Japan never adopted Germany's overall rejection of and contrition for their WWII actions. For example, there's not an annual question of Germany's Chancellor paying respects at the graves of Nazi war criminals. The U.S.'s defense treaty with Japan is a cornerstone in keeping the region stable and safe - it takes the Japanese offensive military force off the table.

Things are changing though: As nationalism is embraced and fanned by leaders, not just in the U.S. but in Japan, South Korea, and China; and as democracy is de-emphasized as a fundamental right and superior form of government and is treated more like just one alternative among many; the idea that peace and human rights are paramount is being pushed aside in favor of national power. We've seen that movie before, many times, and we know how it ends.

Also, the U.S.'s foreign policy capability and credibility has suffered great blows around the world and in East Asia, under Bush and now under Trump. Can Japan and SK trust the U.S. to protect them - if not, their instability greatly increases. They must trust that the U.S. both decides to fulfill its promises (as the U.S. threatens to or does fail to meet its promises under other treaties and promises, from NATO to Paris to Iran to Syria to NAFTA to multilateral institutions), and to capably, effectively execute a major war (Congress' sequester has badly reduced military capability, Congress is unable to act, the Commander-in-Chief doesn't inspire confidence in an essential role, and the military's failures in low-end wars in Iraq and Afghanistan makes it look limited)? Also, the role of the U.S. is to provide stability and to support democracy and liberty, but those don't appear to be priorities for Bush and especially for Trump.

Some stuxnet type hack that makes the next test missile travel west might be interesting.

Do you mean let South Korea/Japan build nuclear weapons or start a war that wipes them off the earth?

I'm certain Japan (the people and even the government) would be against this and you'd wonder if China would start thinking preemptively if they were to start.