I think the problem a non-nuclear WW III scenario is that for a “WW 2.1” style war, you’d need countries that are industrialized, technologically advanced, international “power houses,” but almost all the powerful, technologically advanced, “Major” countries (i.e. The U.S., U.K., Russia, China, France, India, Israel, etc.) have nukes, and thus are deterred from attacking each other. (M.A.D., and all that)
Most of the industrialized but non-nuclear “secondary” powers are allied with a superpower already, who’d probably do a lot of the fighting in place of the secondary power anyway, if the situation was bad enough. For example, say Indonesia decides to attack Australia. Now, if Australia was left all by herself for some reason, she could probably defeat Indonesia in time. But if Australia calls up it’s “big brother” the U.S., we’d probably send over a carrier group or two to break Jakarta over Uncle Sam’s knee, within a matter of weeks. Like taking “a shotgun to a knife fight.”
Realistically, the only “Major: Conventional” war scenario I could see would be something like a “Falklands II,” with a politically “isolated” major power going to war against an unaffiliated semi-industrial power.
Or, perhaps, a much more unlikely scenario causing NEW powers to emerge…say, some sort of Neo-Fascist “Axis” arises between Nigeria and South Africa, begins to swallow up most of Africa, and Kenya and Egypt ally with each other against the Axis. But hell, even then, the rest of the world would more than likely intervene militarily long before things ever got that bad.