I'm interested to know who would have been responsible for policing and investigating crimes committed by Allied soldiers occupying Japan between 1945-1952? Would local Japanese police have had any jurisdiction?
The BCOF, SCAP? I haven't been able to find much in the way of their organisational structure.
The armed forces are subject to the jurisdiction of military justice, not local authorities. US military offenders were subject to court-martial under the Articles of War, which was replaced by the Uniform Code of Military Justice in 1951. It's the same for soldiers who commit crimes on US soil, while a soldier is in uniform he is under the jurisdiction of the military so it doesn't matter where he breaks the law.
The Articles/Code was and still is enforced by the military police corps. If a soldier was arrested by local Japanese police, they would have to turn him over to the MPs for trial as they had no jurisdiction over the US military.
Sources:
-Uniform Code of Military Justice: http://www.au.af.mil/au/awc/awcgate/ucmj.htm
-A Manual for Courts-Martial, US Army, 1928: https://www.loc.gov/rr/frd/Military_Law/pdf/manual-1943.pdf
-Robert K. Wright, Military Police: Army Lineage Series, Wasington D.C., United States Army, 1992: https://history.army.mil/html/books/060/60-9-1/cmhPub_60-9-1.pdf