The short answer is that it took until the 20th century for man to develop the means to travel to and wage war across the large distances required to be considered a 'world war'. Likewise it took societies that long to organize and industrialize themselves to such a scale.

Stone age Egyptians could form all the alliances they wanted - they had no means of contact with or even knowledge of stone age Chinese or stone age Native Americans.

A more complex answer is that the First and Second World Wars are only called such because that's historiography. The Napoleonic Wars have almost as much right to be called 'The First World War' as the conflict between the Triple Entente and Central Powers if you go by relative scale and geographic extent of the conflict.

However, World War One and Two stand out from any other war because they mark significant turning points and shifts in European history and outlook. I believe that the World Wars obtained such significance (and grandiose names) not just because of the scale of battle but because of the political and economic implications that they carried as well.

I'll try to add a little to the already solid answers in this thread.

In general, it's only in the 20th century that communications, economic, and industrial levels achieved a state where conflicts spanning the globe became much more feasible. Nevertheless, "world wars" prior to the properly named ones did exist earlier than the 20th century. The 7 years war, the American Revolution, and the Napoleonic wars each have strong claims to being "world wars" as they were indeed all wars of global breadth with battles across different continents and oceans, and with geographically far separated combatants. And, too, even earlier conflicts might be properly categorized as global conflicts depending on the criteria involved.

However, the term "World War" as it's used today is a term packed with levels of meaning, and not simply a synonym for "globally spanning warfare". World War does indeed mean intercontinental, globe spanning war as a first step. But it also means "total war". It means a very different level of occupying military forces and a very different level of order of battle. The battlefield in a world war could be the size of an entire country. And those trapped within, civilians included, could be subject to starvation, gas attacks, strategic bombing, shelling, etc. Armies would be called up from mass conscription. And the entire economy of a nation or even an empire would be put on a war footing and bent toward prosecuting the war. In a world war it's not simply a matter of a power being at war, the war is the primary driver of day to day life even on the home front.

In a World War you might have a battlefield that is hundreds of kilometers in dimension and battles that last for months with local forces in the hundreds of thousands or more, and casualties in the tens to hundreds of thousands. Such as the Battle of the Somme, the Battle of Stalingrad, the Normandy Invasions, the Battle of Okinawa, and so on. The sheer scale, and the necessity for such huge armies to be supplied via a significant logistical supply chain defines that kind of warfare.

Moreover, in a "World War" it's very, very difficult to not be a party to the war, and nearly impossible to escape the war. Nearly every country on Earth would be involved, and even neutrality would not be a guarantee you could escape the consequences of the war, or indeed invasion and occupation by one of the major coalitions (as was the case with Norway during WWII).

In short, if you lived on Earth during WWI or WWII it's very likely that those wars had a significant impact on your life, no matter where you were or what you were doing. When you look at a map of belligerents and the effective allies or participants in WWII it's pretty nearly every country on Earth (for example, the Swiss manufactured weapons for the Axis, the Spanish were effectively non-combatant allies of the Axis, etc.)

Nevertheless, even redefining "world war" in this way it's still fairly clear that the Napoleonic Wars should still qualify. The battles were unquestionably of a global reach and of massive scope (armies of hundreds of thousands meeting in the field), and very much "total" though thankfully with less loss of life than in the wars of the 20th century. But it was the industrialization of war combined with mass conscription which made it not just possible but in a sense routine for massive, nearly existential, total wars to be fought across the globe in the 20th century. What the Napoleonic wars dabbled in the World Wars took up as stock and trade. It became routine for hundreds of thousands of civilians to be slaughtered either through intentional genocide, wanton murder, or rank inhumanity toward the enemy (as with the Rape of Nanking, the holocaust, and the starvation of PoWs and civilians by both sides on the Eastern front throughout WWII) or through attempts to strategically destroy or degrade industrial capacity and the ability conduct war (as in the interdiction of shipping and the systematic strategic bombing of Japanese cities during WWII). Unprecedentedly large battles and operations would be topped by yet larger battles throughout the course of the wars (the D-Day invasion, the Battle of Kursk, Verdun, Somme, Spring Offensive, etc.)

And instead of merely being enormously in debt and having lost some territory after the war countries had to content with utterly devastated economies, burned out cities, and entire generations that felt the impact of the war (with huge percentages of loses and even more wounded and traumatized, including civilians and conscripted soldiers).