-The problem with nukes is that is a big boy club, Russia, the US , China, France. you have noobs like North Korea or Israel but is more like a big boy thing. One the requisite of joining the big boy nuke club is that once you have nukes you have to learn to live with the fear of getting hit by nukes, like India and Pakistan or Russia and the US, China and the US. Being the country that US used to scare the Soviets with their nukes I don't think that the Japanese public have the appetite of living with that fear.
-Nuclear weapons are costly, being in the big boy club doesn't come cheap. So for Japan a country that is struggling to meet their defense spending, the average right wing Japanese politicians have to tell to the average Japanese voter that people pensions and welfare programs are going to be sacrificed in other to pursuit a nuclear arms race in Asia STARTED BY their own clownish right wing politicians OR sacrifice conventional weapons programs.
-Another problem, is that you have to design a nuke, even if the US give Japan a nuke design, you have test your nuke to make sure is going to work in a real war scenario like the Big Boys in the Nuke club did, Supercomputers and lasers has their limits. If Japan test a nuke, apart from ruining Japan reputation, it will send China nuclear weapons program in overdrive, to the South Koreans to build nukes, the North Koreans to built more and bigger nukes and worse of at all for Japan to the Russians to move part of their arsenal to target Japan.
-A nuclear Japan it will be seen as immediate threat by China, It probably send China nuclear program in overdrive probably forcing them to reach nuclear parity with the US in a short time, the US would not be happy about that. China nuclear arsenal and doctrine has being designed to deal with the US not Japan and giving the closeness of Japan to China it could force the country to abandon their "non first use" doctrine so in a emergency they could take enemy nukes and cripple a country before is used on them.
-The Japanese are an ally of the US, that is a big problem for Russia. Is almost sure that a nuclear Japan would be forced to attack Russia. So in the case of a Nuclear Exchange between the US and Russia EVEN IF Japan is not involved in the war, just by the fact that is an ally of the US with nukes is almost sure that the Russians are going to try to severely cripple the country. Pretty much like they would likely do with against the UK and France.
-That leave us with the biggest problem that Japan has for nukes, space, just by seeing a map of Japan it seems that there isn't a single place in that country that you can put a nuke or storage nukes that when its get hit a city and inhabitants are vaporize in a instant. The average nuke comes in three nice flavors, Landbase (Silos and Mobile), Air based (Bombers), Sea Based (Submarines). Everything in land could be hit, Air bases, storage facilities, silos and TEL. The best option would be only submarines but that would be extremely costly compared to Land base options and would severely limit the size of any nuclear arsenal compared to land base. Also would create a submarine cat and mouse arms race with China and Russia.
Yeah, the Big Boy Club. The cool guys with nukes. Joining the club Is costly and come with tradeoffs. Nukes are really expensive and difficult to make weapons and most states would rather use them in extreme cases and not in a conventional war where there are cheaper more abundant options. But that change when you have nukes.
And the situation only get worse when you have bad blood and unsolved issues with your neighbors. The thread of nuclear war will be always be there. The world just tight their butts every single time that the Pakistan and India even insult each other, hopping that these two countries don't commit the biggest genocide act in Human history.
Japan is seen as more or less seen as a threat by China, Russia, North Korean and EVEN South Korea. The country has worse dynamics that than US has with China, Russia and even North Korea, for the US, the dynamic between them and China or Russia is just great power politics, given Japan's past and their right wingers is much worse. So Nukes in Japan would likely change the dynamics for the much much worse.