This is indeed a tough one, with possibly no one right answer.
The links with WW1 are important and accurate, but too much time passed between wars for the end of the 1st to be linked so closely to the beginning of the 2nd. If one wants to view both wars as one period that works, but that would put the start in Aug 1914. (Though the roots falling in the Boxer Rebellion is a good link).
Hostilities began in a minor way in China in 1931, as has been pointed out, but didn't kick off in a very serious way until 1937. Still, this is only a localized theater engagement. The Italian invasion of Abyssinia is also a beginning of hostilities of some sort, but had no real significance to the overall conflict.
Europe and the colonies becoming enmeshed in the conflict in 1939 does bring participants from all hemispheres, but they come to fight in Europe. The Europeans didn't get involved in fighting in Asia until later after Pearl Harbor (I think, someone please check me on this). As such this is it's own globally participated theater engagement. A counterpoint to this is the alliance of Germany and Japan. If one sees the World War as Allies Vs Axis, this that been globally satisfied at this time.
The German attack on Russia brings not only a new front to the European theater, but does span continents. Remember that the USSR extends well into Asia. While the fighting didn't really leave the European part of it, an important part of the Russian war effort was moving production east and out of range of German bombers. This could be called and extension of the conflict to global proportions.
The attacks on Pearl Harbor, the Japanese drive towards SE Asia and the declarations by Germany and Italy that followed did fully bring the conflict into global status. In this case America is the bridge that spans both the oceans, and is the only combatant that is fully committed to two theaters of war (counting Africa as Europe for the sake of geographical simplicity).
Precursors to the war such as the remilitarization of the Rhineland and the takeover of the Sudetenland are important to consider in the causes of the war, but did not directly lead to hostilities, and as such I don't think they can be considered the beginning.
I think the best single answer is the beginning of Operation Barbarossa, opening the Eastern theater, brining Asia into the fight and eventually leading to the defeat of Germany. This was the single biggest front of the war, and was probably most responsible for brining it to a close. Without the strain of the Eastern Front, the tides of war in the West could have turned much differently. The entry of the US into battle in Europe and the Pacific later in 1941 seals the deal, conveniently just as things start to go bad for the Germans in Russia, and represents the high water mark of the Nazi war effort. Still, I think that there was enough conflict across the world by that time to consider it a world conflict before the entry of America.
I guess it all depends on how you define a world war.
