I would like to expand on the last paragraph of Thucydides' already excellent answer. I hope this will allow you to understand better that the use of Nuclear Weapon in the next 'global war' is an unlikely as ever.
The Russian operations in the Ukraine is a great example of how wars will be in the future. The days of countries openly and formally declaring war on each other and then proceed to prosecute that war with all the limitations imposed on the belligerents by any Laws of War such as the Geneva Conventions or things like the Law of Land Warfare is fast coming to an end. This is mostly due to nation states' realizations that they have a lot more strategic and tactical flexibility if they engage in an undeclared hybrid war such as the one the Russian Federation is prosecuting in the Ukraine, Iran is prosecuting in Syria, and others. (For a comprehensive study of how Russia prosecutes this war please read this highly interesting paper by the Potomac Foundation.)
'Hybrid Warfare' or 'Non-Linear Warfare' or the Russian 'New Generation Warfare' is of course a term first popularized by GEN Valery Gerasimov, the CoS of the Russian Armed Forces in a paper titled The Value of Science in Prediction(linked article in Russian. Here's a good translation.) According to Gerasimov:
In the 21st century we have seen a tendency toward blurring the lines
between the states of war and peace. Wars are no longer declared and,
having begun, proceed according to an unfamiliar template.
He then continued to press his point in the most interesting, and important, paragraph of the article:
The role of nonmilitary means of achieving political and strategic
goals has grown, and, in many cases, they have exceeded the power of
force of weapons in their effectiveness.All this is supplemented by military means of a concealed character, including carrying out actions of informational conflict and the actions of special-operations forces. The open use of forces — often under the guise of peacekeeping and crisis regulation — is resorted to only at a certain stage, primarily for the achievement of final success in the conflict.
This means everything from information warfare (agit/prop), the use of proxies (rebel groups, militia, irregular forces), the use of undeclared SOF (see Little Green Men), economic warfare, the intermixing of GPF/SOF personnel in proxy groups (declared as 'volunteers'), the support of proxy groups by high-end kinetic capabilities (cross border artillery, UAVs, etc).
The war in the Ukraine, Iraq, and Syria are nasty conflicts without a doubt. Anything from massed fire into civilian areas to the summary execution of wounded combatants were seen. What we do not see, however, is the use of nuclear weapons.
There are basically four reasons why your country needs to deploy nuclear weapons:
- To destroy concentration of forces. This is what NATO planned to do against massed Soviet armored formations if they ever storm the Fulda Gap.
- To target the enemy's nuclear weapons facilities (the First Strike use to ensure elimination, or at least severe reduction of the enemy's capability to retaliate)
- To target enemy cities. This was envisioned to destroy the enemy's
will to fight. This is no longer applicable in the 21st century. No
nation-state will ever deliberately target an adversary civilian
population with nuclear weapons. Hell, not even with chem/bio
weapons. The Syrian use of chlorine gas against rebel positions that
happened to have civilians in it was a very different ball game to
the deliberate US targeting of Hiroshima and Nagasaki.
- And most importantly, based on the three above, to deter adversary aggression based on the threat of force.
All these are nice and good if you're engaged in an open, conventional war with an adversary state, where combatants of both sides wear uniforms, acting under legal orders, and are bound by the Geneva Conventions. In a New Generation War? Not so much.
The use of nuclear weapons are always a double edged sword. Yes, its massive destructive power will obliterate anything you hit. But on the other hand, anything it hits is no longer viable for extended operations. Even if your armies can operate in MOPP, they would not be able to prosecute effective combat operations for long. Their MOPP suits will quickly degrade their performance, use up supplies, and make the area unsuitable for staging.
Moreover, it will alienate the using power from the population it seeks to either win over or cow, quickly limiting its ability to operate freely in its AO, making your armies devote too much time to Force Protection instead of offensive operations.
It will also break the shield of deniability that the user has projected to hide their operations. Russia can deny the Little Green Men were theirs (until, of course, they admitted it), but if all of a sudden accurate, massed nuclear fire hit Ukrainian Army positions - well, you can't hide and deny that.
So to answer your question: The way WW3 happens on your world without the use of nuclear weapons is to realistically portray the conflict as a series of interconnected low-intensity hybrid warfare prosecuted by several different governments, their irregular proxies, and various other non-state actors (Al Qaida, IS/Daesh, Boko Haram, Abu Sayyaf, etc) that take place around the globe at roughly the same time for a decade or even more. Its close enough to what we are experiencing right now that you can make the scenario very realistic.