Rebuilding efforts were funded by the occupation government, which pumped a lot of money into Hiroshima and Nagasaki. Acts of war are usually not covered in home/business insurance policies, so I doubt that insurance companies had much to do with any of the rebuilding, though I may be wrong. Large sections of Tokyo and other cities were destroyed by bombings and fires, so if insurance companies were forced to pay for all of that property damage, they'd all go bankrupt.

Initially the hibakusha (survivors of the atomic bombings) were treated through funds and supplies by the wartime Japanese government and then by the occupation government. Then they were given free national health care policies in 1957. However, for almost 20 years, this did not extend to non-Japanese survivors of the bombings, like the many Koreans who were in Hiroshima at the time (forced labor, migrant workers, etc).

As of 2007, there were "226,598 officially certified survivors" of the atomic bombings still living in Japan who get monthly reparations, with some getting additional medical stipends. It sounds like you've read about this a bit, so you probably know that there was and still is a stigma attached to the hibakusha. So in those cases, a monthly stipend doesn't correct things like workplace/hiring discrimination.