I'd also welcome pointers to source material and any interesting stories you may have come across about military police in this period. Thanks!
I cannot comment on the US MPs or Shore Patrol, but as for the British Corps of Military Police I can give some infromation.
Nowadays the "Redcaps" as they are affectionately known belong to the Royal Corps of Military Police, but before WWII they were simply the Corps of Military Police. The Structure of the Military Police is very important to note, we all have images of Redcaps walking into bars and throwing rowdy troopers into the backs of vans and land rovers and carting them off to a cell but the military police were not usually the ones who were involved in these situations.
The two main organisations in the British Army who are related to the law are the RCMP/CMP and the Regimental Provost Staff. The MPs are their own Regiment and wherever the Army goes they will follow, but the "Provos" are each attached to their 'mother' regiment and are pooled from the regiments own soldiers.
Provos are distinguished by a black armband worn on the right arm, or a DZ flash with the initials RP. They have very few powers that are essentially limited to making sure that soldiers in their regiment or corps don't get themselves into trouble. Provos do not have any special policing powers and do not have the ability to launch any form of criminal investigations, but they hold the rank of Corporal or above which gives them rank over most of the men by default, they also have power of authority due to their role.
The CMPs were the men who would handle any criminal investigations. During the second world war and even afterwards they had a lot of legal power. Soldiers could spend their entire lives in stockades for crimes such as murder, nowadays the maximum sentence for a British soldier is 6 months! This is of course due to the fact that in todays army all major criminal offences are handled by the RCMP and the local police, so sentencing is carried out in a court of law and they are sent to a civilian jail.
In the second world war the CMP operated in a variety of roles, from guarding static positions such as bases ,supply depots and PoW camps (Which is now the role of the Provost Guard Service) to investigating war crimes and serious criminal offences such as murder and rape.
Hopefully this gives you a little bit of insight and helps you out, I apologise if it's not what you are looing for.
You might be interested in the Battle Of Brisbane - the largest combat to take place on the Australian mainland (apart from the Emu War).
http://en.m.wikipedia.org/wiki/Battle_of_Brisbane
A huge riot during WW2, partly started because some Australian troops didn't like the way some American MPs were treating an American soldier.