I know there are questions pertaining to "What would happen if all nukes were exploded at X location?", however, I can't seem to find a question asking this: What would happen if all of the Earth's nuclear weapons simultaneously exploded where they are now?
This fictional scenario may be less catastrophic than the one cited where the weapons are used to maximum destructive effect in a nuclear strike and retaliation. If they detonate where they are stored, that means underwater or underground in a smaller number of locations.
Unless they all go precisely simultaneously, the first to go will destroy the other warheads in its vicinity without exploding them. Also the yield of a warhead depends on a precisely timed sequence. Any departure from this will reduce the yield, quite possibly to a mere few tons ( a "fizzle").
Even in the worst case where the explosive yield is the designed maximum possible, the number of fireball locations will be far smaller than with the actual use of the weapons in anger. So there will be less consequential fires and less soot induced global cooling.
For an underwater detonation there will be no fire. Most of the energy will vaporize water which will soon condense and fall as rain. For an underground detonation a greater amount of energy will be absorbed by rock.
There is some bad news. The fallout will be terrible. An underground burst will be worse than a ground burst. (I am assuming they are not buried deep enough to contain the fireballs). Most if not all of the bombs are in the Northern hemisphere and global air circulation tends to delay mixing between hemispheres. Fallout radiation decays rapidly. New Zealand will be least worst affected?
Putting these various things together ought to allow you enough convincing handwavium to justify whatever degree of disaster your plot requires.
Andrei has it right, but I'm not certain about any tectonic problems. Ploughshares.Org cites that at this time there are 15,375 nuclear weapons remaining in the world, a number down by almost 60% since the peak proliferation in the 1980s. While reduced, the number remaining is far in excess of what would be needed to end life as we understand it.
Your question poses them exploding simultaneously. While this may or may not be possible, the resulting devastation would definitely include the following, as quoted from "Bulletin of the Atomic Scientists", 1984. With the 'worst case' scenario, and detonation of 16,000 weapons with approximately 10,000 total megatons of energy released, 63% of those as surface bursts, the resulting particulates, smoke, soot and airborne fallout would reduce ground level temperatures to approximately -50 C for a year or more.
The initial detonations would reduce the population by 2 to 3 billion as urban areas are hit and irradiated, burned or vaporized. The subsequent collapse of infrastructure in the following hours would increase that toll by another 1-2 billion as over the first week, untreated radiation sickness, thirst and starvation settle in with near complete darkness. The remaining populations' survival rate depend on a great many variables, chief among them, location and proximity to shielded space to protect from cold, radiation and disease.
I highly recommend "Bulletin of the Atomic Scientists", 1984, available at Amazon, it answers in scholarly, scientific depth the exact details of many such scenarios.
As you might expect, there are no detonation scenarios where life just goes on...