[quote=@Unfortunately]
I suppose the definition of a WW3 here is the full-blown usage of nuclear weapons and other similar weapons of mass destruction that would turn this into a mutant-less Fallout NRP, so the drastic event would be conventional warfare with a scale large enough to have a significant impact on human civilization.

Alright. That's okay, I guess.
[/quote]The effect of nukes are somewhat overblown. We never came close to possess enough nuclear weapons to literally wipe out ourselves. At worst it'd have some negative effect on the environment and detonation zones would be like Hiroshima but otherwise?
Wrecking the planet or its ecosystem? No. It would be nasty but the planet had much worse.
Killing all humans? Not even close even if you include radiation fallout.
Nukes are also fail-safe which means destroying a nuclear missile won't trigger the reaction so intercepting nukes is rather safe in general.
Interception would of course reduce successful nuclear strikes considerably.
Nukes are terribly powerful but I often feel people are blowing their threat out of proportion.

Okay, excuse me my rant. It just bothers me that everybody treats nuclear war as it would truly result in the end of the world. Nope, even Fallout-like dystopias are unlikely. 

Back to the point, sorry I misread your OP and thought your game would feature space travel and such from the beginning.
If you want a setting where we struggle for the "black gold" then 2060 is about right.
and nah, ww3 is indeed not necessary here, then.
Albeit the closeness of the timeline means nations would be mostly similar to what we have now. If you want to avoid this you have only one method: Alternate Universe
So UK exists in our world? Not this one! Welcome with loud cheers the Gaelic Federation!
Do you wish for an Egyptian nation which still believes in the ancient pantheon? You got it!
Alternate timeline would allow people to diverge from present politics and history by creating their nation from almost a scratch!
Maybe this would be the best solution you can get.
On the other hand this pretty much means you can just tke my tech suggestions from the Age of Rebirth 2 game. That should work as a starter.
If you wish to make people struggle for oil more, you may wish to nerf on alternative energy sources. Fusion powerplants may solve much or dependency on fossils for electricity. And advanced power sources may dodge the issue with the shortage of oil. So perhaps you limit these?
If explanation is needed the date could be also moved to the 2030s. Some predictions say that this is the point where the extraction of oil would have questionable economic gain.
With closer date technology will be less advanced and people can dodge the oil crisis less.
It all depends on what kind of game you wish to make.