Was it the War of Spanish Succession that was the first war to occur on three continents. If that's not a World War, I don't know what is.
Now as for trenches and mustard gas...
I would have thought the ancient world holds the claim for the first war to occur on three continents... For example, the Octavian-Antonine civil war that ended the Roman Republic touched on Asia (Minor), Africa (Egypt) and Europe (Greece).