It was a combination of several things. In Germany it was usually tasked to civilians, military, and, if at all available, POWs. Fire bombs and heavy bombing raids were particularly messy. There were casualties in every raid, usually civilians caught in the crossfire. It has been argued as to whether, the allies dropped warning leaflets or not. I would like to believe they did. Neighboring civilians or those left from the bombings usually did the cleaning, military helped mop up as best as they could. Sometimes a train would stop and POWs, most notably American, would be forced out to help with "their" mess. Many were beaten, spit on, and in some cases killed by the mobs of civilians that saw an enemy they could kill on the ground and not in the sky. Most often, civilians would bury the dead in nearby graves. In the case of the Battle of Britain, however, Auxiliary organizations were created in the run-up to the battle. Each neighborhood had an air-raid warden, which commanded locals who held specific jobs, often that matched their day jobs. Nurses turned into stretcher-bearers, clerks and secretary's into report and control, boy scouts into messengers, etc. Air raid bases were often set up into churches, homes, shops, the like and was usually a local affair. Sort of like a neighborhood watch, but obviously much more serious. The result of german bombings for them were similar, they formed rescue parties, if needed first aid crews would be on-scene and ambulance drivers would take them to the nearest hospital. The dead were logged in notebooks, reported and handed over to the authorities to be buried. Survivors of these bombings often were placed in the care of local powers, the church would let them sleep while they looked for a relative, relocation would be helped, some neighbors would put you up in their home. It really varies from case to case, you would need to look at specific bombings to get a handle on the aftermath, but you get the idea.

Sources: The Last Mission of the Wham Bam Boys: Courage, Tragedy, and Justice in World War II by Gregory A. Freeman Citizens of London: The Americans Who Stood with Britain in Its Darkest, Finest Hour by Lynne Olson