Why World War III?

Good question. I assume since it was a highly secret project experimenting on children, they covered it up and wouldn't admit to something that killed thousands. Also, in nuclear war, quick response is the best strategy.

First off, Akira and the three Espers we see in the film were created in a top secret government experiment to try and develop psychic powers. Unfortunately for the scientists in charge of this experiment, Akira gained too much power too quickly and achieved Godhood before they could administer drugs designed to reign in such powers(the same drugs the Espers have been taking for decades).

Because, you see, in the Akira universe, psychic powers are the same kind of powers that God(the ability to create and manipulate your own reality) would have and the implosion at the beginning of the movie is of Akira's power becoming so great it basically collapses in on itself (just like a black hole is created when a star's gravity becomes too great and it subsequently collapse in on itself) thus creating a singularity which in turns gives birth to a pocket universe(with Akira becoming its creator God essentially). Basically, Katsuhiro Otomo has mixed the tenets of Buddhism with quantum physics.

Now, the Japanese government but not the rest of the world is privy to these experiments. So, to an outsider, the implosion would appear to be just like an atomic bomb attack and, what with it happening in 1988(not 1992 as another poster said with 1988 being right at the tail end of the Cold War) someone was paranoid enough to give in to their "itchy trigger finger" and let off their own stash of nuclear weapons thinking that a nuclear war had started. Other nuclear powers followed suit(ah, the joys of (M)utually (A)ssured (D)estruction) eventually developing into a full-blown WWIII.

So basically, the movie is about the government trying to start up the experiments again using Tetsuo. Unfortunately, just like their predecessors, their arrogance leads them to think they could control something they couldn't resulting in another Akira-like person to develop(Tetsuo). Tetsuo, in turn, looses control of his power by the end of the film, said power collapses in on itself creating another pocket universe and Tetsuo becomes the creator God of it.

Does this clear up your confusion?

One day men will look back and say that I gave birth to the twentieth century.

So how where the scientists able to recover Akira's body, but Tetsuo is nowhere to be found? Was NeoTokyo actually self contained in the Akira incident, and the ball of light that represents Tetsuo actually a self contained universe? The movie makes allegories that a creation of a universe takes place at the end.

A slightly different theory:

Large explosion in Japan

World: What was that?
Japan (needing to cover-up experiments): Oh nothing, don't worry about it.
World: It's quite concerning, we really need to know what's going on.
Rhodey (on behalf of japan): Unfortunate training exercise.
World: We're going to need more detail than that, a city's been destroyed.
Japan: Seriously, it's nothing.
World: You have to tell us, if not we're going to find out ourselves.
Japan: Please drop it.
World: Send in the troops.
Japan (to other countries likely running their own experiments and supportive of a cover-up): Little help?