What did the Japanese medical response to Hiroshima and Nagasaki look like? How long did it take to figure out they were dealing with radiation injuries?

For a gripping first-hand account of exactly this, I recommend Hiroshima Diary by Dr. Michihiko Hachiya. Hachiya was the director of a private company hospital in Hiroshima at the time of the bombing, and his diary covers everything from the day of the bombing to the arrival of American troops in the city weeks later, with extensive speculation written in the moment about what we know to be radiation poisoning, and the process of coming to recognize what symptoms tended to lead to survival versus death. There is mention towards the later stages, several weeks after the bomb, of early suggestions and then understanding that some of the injuries seen are radiation injuries, but that had limited affect on the response given, due to the utter destruction of most capabilities, and the limited availability of supplies or places to use them.

The treatment given generally consisted of treatment for burns, attempts at hydration, and the application of clean bandages when available. He observes that proper nutrition appears helpful in the recovery of the less severe cases, and does his best to coordinate the medical staff, triage process, patients, and the piles of random family members and civilians who descended on his hospital. He does this all while himself recovering from terrible burns, blown-out ear drums, and numerous injuries caused from his house collapsing on top of him.

There is a certain surreality about it, as he gets better and begins exploring the blasted city seeking help or information for his patients and himself. He describes a mass of people surviving together in the pitch-black concrete hulk of a shopping center in conditions that are eerily similar to descriptions I've heard of the Superdome after Hurricane Katrina, and at one point is able to make it to a train service that returned to operation, leaves town to visit a friend and have tea at an upscale restaurant, and then returns to his blasted-out hospital to continue caring for people.

The medical system in Hiroshima was not a network of public hospitals that all worked together and helped each other out, but rather a series of private hospitals for company employees (like his), military facilities, clinics, etc, and each seemed to look after itself. He has to argue with Japanese military officers about supplies, with some attempting to requisition his supplies for themselves, and others willing to give him theirs to help care for civilians. A series of islands in a storm, each governed by its own set of rules, working with the knowledge available to the people on hand, seems to have been the rule. I particularly like this book because it was not written to be published - it is a genuine diary, which was only learned of and translated several years after the war.