The Wikipedia entry for World War II uses the start date of 1 September 1939 because Wiki is applying the Wikipedia definition of the term "world war" as a basis, for internal consistency. One must start with a definition of that term to formulate an answer as to why.
Many good answers here already - each with valid points to consider, and any one of which I could accept if I had posted the question (and a +1 from me on more than one of them). However, the definition of World War has not yet been addressed (i.e. when is it appropriate to use the term World War in connection with an international conflict?).
The various definitions of the term itself include no objective criteria (e.g. 60% of nations of the globe, or 60% of the land-mass of the globe, etc.) but rather subjective criteria. For example, if using Merriam-Webster's definition (my emphasis added):
a war engaged in by all or most of the principal nations of the world
... one must first decide what constitutes a principal nation, and then, if they are not all engaged in finding ways to kill each other, we move to the next qualifier, most, and then try to decide how many is "most"? 50.00001% or more? Does most require a two-thirds majority? Three-quarters?
With such subjective measures and no objective qualifications, one then turns to historical usage of the term to find a less subjective answer. Who first uses the term, and when? Are those sources considered authoritative?
From the Wikipedia entry on World War, there is much to be learned about the historical use of the term, which may provide an acceptable (if not definitive) answer to this question. It comes down to this: When does an international conflict become a World War? Definition from Wiki:
A world war is a large-scale war involving many of the countries of the world or many of the most powerful and populous ones. World wars span multiple countries on multiple continents, with battles fought in many theaters. While a variety of global conflicts have been subjectively deemed "world wars", such as the Cold War and the War on Terror, the term is widely and generally accepted only as it is retrospectively applied to two major international conflicts that occurred during the 20th century: World War I (1914–1918) and World War II (1939–1945).
Origin of the term (also from the same Wiki, my emphasis added below):
The Oxford English Dictionary cited the first known usage in the English language to a Scottish newspaper: the People's Journal in 1848: "A war among the great powers is now necessarily a world-war." The term "world war" had been used in 1850 by Karl Marx and his associate, Friedrich Engels, in The Class Struggles in France. Rasmus B. Anderson in 1889 described an episode in Teutonic mythology as a “world war” (Swedish: världskrig), justifying this description by a line in an Old Norse epic poem, "Völuspá: folcvig fyrst i heimi" ("The first great war in the world".) German writer August Wilhelm Otto Niemann had used the term "world war" in the title of his anti-British novel, Der Weltkrieg: Deutsche Träume (The World War: German Dreams) in 1904, published in English as The Coming Conquest of England.
In English, the term "First World War" had been used by Charles à Court Repington, as a title for his memoirs (published in 1920); he had noted his discussion on the matter with a Major Johnstone of Harvard University in his diary entry of September 10, 1918.
The term "World War I" was coined by Time magazine on page 28b of its June 12, 1939 issue. In the same article, on page 32, the term "World War II" was first used speculatively to describe the upcoming war. The first use for the actual war came in its issue of September 11, 1939. One week earlier, on September 4, the day after France and the United Kingdom declared war on Germany, the Danish newspaper Kristeligt Dagblad used the term on its front page, saying "The Second World War broke out yesterday at 11 a.m."
The Wiki section on Other global conflicts lists wars which, while global, have not been given the term "world war" (especially World War in capitals, which Merriam-Webster declares to be reserved for WWI and WWII, thus far in history anyway). Why were these other global conflicts not called World War X, Y or Z? Because no one did call them that. And no one does. Though anyone could. It is a socio-political term, not an objective or empirical one. It is a coined phrase. Those who coined it get to shape or influence how it is used. Like D-Day ... every operation prior to 6 June 1944 had a D-Day (day of commencing operations) and an H-Hour (starting hour). After June 6th, the term became synonymous with Operation Overlord and no one thinks of it in any other manner now.
Specific Case: Start of WWII
In reading through the definitions and accepted use of the term, the consensus seems to be that there were global conflicts transpiring before 1 September 1939, but the tipping point to qualify it as a World War (among those who define or shape language or terminology, at least in a popular cultural sense) came when hostilities broke out 1 September 1939 when Germany invaded Poland, which, by treaty, resulted in war being declared by the UK and France (and their respective world-spanning empires, thanks sempaiscuba) on 3 September 1939. Why not use September 3rd then? Because technically a state of war existed upon Germany's invasion of Poland (by way of treaty) 1 September 1939.
As a parallel example of a technical state of war, refer to Roosevelt's Day of Infamy speech given to the US Congress 8 December 1941 (a day after the attack on Pearl Harbor) in which he said in conclusion of his remarks (my emphasis added):
I ask that the Congress declare that since the unprovoked and dastardly attack by Japan on Sunday, December 7, 1941, a state of war has existed between the United States and the Japanese Empire.
The US Congress declared war on Japan 8 December 1941, but that was a legal formality. A state of war existed upon the attack 7 December 1941 (and also by virtue of Japan's declaration of war on the US and UK 7 December, though delivered after the attack started - the attack itself started the state of war). This same logic can be applied to the state of war in Europe between Germany, Poland, the UK and France which commenced 1 September 1939 when hostilities began.
Those who coined the phrase "World War II" applied it to that war which started between those four nations and their global empires when Germany invaded Poland 1 September 1939. And it stuck. And it became accepted as such. It can also be rejected as such.
While Questions that ask "Why..." often invite opinions (and some good ones have been expressed here in some Answers and Comments), this Question can be addressed from a marginally historic perspective, and that is by examining the historical use of the term World War II, when it came into use and by whom, what are its roots in previous usage for other global conflicts (etc.) - hence why I chose to post this Answer from that standpoint. I think this approach can keep the Question grounded in terms that are not primarily opinion-based (even though use of the term may well be challenged as opinion- or politically-based) in hopes that by doing so this Question can remain open for years to come with many more great Answers and Comments added (which at some future date others may refer back to for historical perspective by the way) without fear of having this closed for being opinion-based.