WWIII would never be done justice, and is so close to our time, that any claims would likely insult someone or another - hence why screenwriters also probably kept it as nebulous as possible, only dropping hints about it's nature.
Since oil reserves are predicted to run out around this century, the population of the planet will be 9 billion around 2050 (UN stats), structural unemployment is rising, the environment is growing worse, and we are already using more than the carrying capacity of this planet in terms of feeding our living standards - my interpretation of WW3 has always been a perfect storm; an apocalypse in which all our short sighted decisions, bigotry, over-consumption, and inability to care for each other's needs fairly, as a species, come back to haunt us. Inter-state warfare, terrorism, revolutions, nuclear annihilation, death camps, mass unemployment, proletarian uprisings, nationalist demagogues, religious fanaticism, torture, zero civil liberties, police states, universal conscription.
But the prejudices of our era would seep into the work - Environmentalism, Nationalism, Islamism, Capitalism, Socialism - the energy crisis, the military-industrial complex, consumerism, imperialism - China, India, the USA, the EU, Japan, Russia - the author's own views on these things, may seep into the work - someone would be insulted, in all likelihood - especially if Eastern Coalition = bad guys.
I don't actually think eras should be explored just for the sake of it, for reasons I was recently talking about elsewhere (the desire to explain every gap is destructive IMO, and can often, but if there was an era - purely theoretically - that I wanted to see, it would be WWIII, or the era just after First Contact, or first contact with the Kzinti, or the Tholian war on the Federation, or the Cardassian War. Purely speculatively, I would like to see a longer, more detailed, Romulan War - but really, I don't want to see any of these things, for the reason mentioned above - they were only ever intended as engimatic references anyway.
Trek should probably go into the far future instead, and go back to it's roots, in exploration and awe - at this point, I don't care about the kind of technologies and logistical ideas established in DS9 and Voyager - Star Trek should be creative and awe inspiring - we might as well have wormhole-drives and exploration of other galaxies.