Japan refuses to surrender after the bomb. The typhoons that would have hampered the invasion hit, so the blockade/bombing campaign continues, with the US, through conventional and incendiary bombs, as well as an increasing amount of nuclear bombs, now begins obliterating city after city in Japan, and reduces what's left of the country's transport infrastructure into rubble and attacking vehicles, making travel and food distribution nearly impossible, especially as fuel resources dry up. Chemical weapons that began development in 1944 for the purpose of destroying crops are put into action against Japanese farmland.
By the time the invasion rolls around, Japanese society itself is collapsing. People are desperately trying to scavenge whatever they can eat, often fleeing what's left of the cities to do so, cannibalism is rife and growing, theft is rampant, public services, including law enforcement and the justice system, are collapsing in many places, and law and order is breaking down, as the Japanese government's control over anything outside Tokyo (now an almost completely burned out ruin) is flimsy.
After a massive series of battles which see millions of soldiers and civilians fight to the death, along with a few hundred thousand Allied deaths, mass suicides ensue. By the time the Allies have fully conquered Japan, the state itself has ceased to exist, and the country has been taken back to the 15th century technologically, with people starving by the masses. Soon, the Allies find they have to build everything up from scratch, including an entirely new Japanese government, as the political and military leadership is not there.
The occupation manages to stabilize things and bring the humanitarian catastrophe to an end, but now, there is no Japanese government ruling alongside the Allied occupation authorities in OTL, just the occupation authorities. In 1948, after the situation has been stabilized, the Allies face a rebellion over the execution of Emperor Hirohito. During the invasion, the Japanese had carried out their threat to slaughter all approximately 100,000 Allied POWs in their captivity the second it began, leading to the further erosion of goodwill, and Hirohito ends up on trial. This results in isolated acts of violence, leading to a crackdown and the erosion of further goodwill.
With the Cold War on, the US builds up Japan, and it slowly recovers. The US military begins recruiting Japanese, and many young men, who see it as an opportunity for a better life, sign on. Japan also becomes an important air and naval base for the US during the Cold War.
The US authorities oversee the educational system to wipe out the fanaticism of the Imperial era, with American teachers brought in, and young Japanese grow up familiar with American culture and speaking English. By the 1960s, with the occupation going past it's 20th year, the "Japan question" is on everyone's minds. The US is aware it can't keep a population under permanent US rule without any political rights forever, and the primary consideration is Japanese independence with a treaty stipulating that the US will be permitted to keep basing large military forces in Japan. While the older generations would like to see the Americans gone, a movement arises among the younger, highly Americanized generation for further integration, especially considering the massive prosperity of the United States relative to the rest of the world at the time, in addition to improving race relations. Demonstrations alternatively demanding independence or the vote are common. The ball is pushed further and further forward until 1980, when a referendum results in Japanese voting for statehood. Japan becomes a US state the following year.
Almost totally ASB, but there you go.