I've been looking at other answers, but I'd like to know, science
based, if anyone could survive a nuclear war and all that follows.
Could anyone survive? Yes. Across the United States there are a variety of fallout shelters. Some are public, most are private. A few are for government use and hence, not publicized. Add to that, fallout shelters in other countries, like Canada. So, assuming people get to the existing shelters and they are not in the blast radius of a weapon, yes, some people could survive a nuclear war. How long they survive after the blast depends on the amount of supplies they have.
I've read many things saying they would, and read other science-based
answers, but they all talked about a nuclear war that was relatively
small (100 nukes fired between Israel and Palestine). Would anyone
survive a nuclear war of NATO vs. China and Russia, or something
similar, where all the nuclear bombs they could fire were launched?
There are estimated to be over 15,000 nuclear weapons held by the nine countries known to have such weapons. About 2,000 of those are on a "High Alert Status" meaning they are already programed with a target and could be deployed (launched) in minutes. Russia and the U.S. account for about 14,000 weapons and about 1,800 of the "High Alert Status" weapons.
So, let us say all 2,000 that are on "hair triggers" are fired. While we don't know the exact yield, a fair estimate for a modern weapon would 1 megaton each. Due to the wide range of devices and delivery systems, setting an average is not simple. Each five megatonnes of explosive power translates to about one ton of mass for the weapon. A 1 megaton device would clock in under a quarter ton. That is a good size, but manageable for several delivery systems, both short and long range. While we have tested much larger yielding devices (15MT, 25MT, 50MT), smaller is more practical. Also, as a device's yield grows, its destructive ability only matches that growth to a point. The reason is the earth's atmosphere.
It is the pressure of the atmosphere that spreads out a blast. As you go higher, the pressure of the atmosphere is less. This affects the nuclear blast in that it, like any wave force, will follow the path of least resistance. So, as the power of the blast increases, the area of planet's surface affect will not increase in proportion because the lower density of the upper atmosphere, having less resistance than atmosphere near the ground, directs the blast upward more than outward.
Indeed, the difference between 50 and 100 megatonnes at ground level would be very small because, even with twice the power, the 100 megaton blast would lose most of its power from the escape of energy to space as it would blast all the way out of our atmosphere. At least that's the theory based on the biggest test ever.
Assuming 1 megaton each, that means a total force for all 2,000 would be 2,000 megatonnes (2 BILLION tonnes of TNT). That's enough force to throw millions of tonnes of debris high into the atmosphere. Add to that all the ash and smoke from the fires caused by weapon detonations, easily adding millions more tonnes of radioactive material in the air. These particulates blocking sunlight, and the fact that they are a radioactive, is what gives us the term "nuclear winter". That debris would remain in the atmosphere until it settled back to the ground. Rain would be the biggest player in that process. Wind and rain would carry this radioactive fallout across much of the planet's surface.
No one knows, for sure, how long it would take that much matter to rain back to earth, but a few years is easily possible. During that time, sunlight reaching the surface would be reduced. This would affect the earth's ecosystems, as plants die or are reduced in number, the higher organisms would suffer as well. On top of that, it would be colder with less sunlight.
Of course the air clearing is both curse and blessing. While clearing air means more sunlight and heat it also means that some areas get hit by the radioactive rain. Such places will be contaminated by fallout for many years, anyone in these areas will most likely die, some faster than others.
I've read things would be better off in the Southern hemisphere, but
how would they be, and how many people would survive?
This is due to the fact that most targets for nuclear attacks are in the northern hemisphere, hence, that's where the firestorms and ash clouds will originate. Weather patterns between the north and south are such that most of the airborne ash will remain north of the equator. That's, in part, due to the sun's effect on air near the equator, it's hotter there so more air is pulled in from the colder areas near the ground and, likewise, expelled toward the colder areas at higher elevations. While this air current wouldn't stop ash traveling to the southern hemisphere, it would reduce it.
That translates into less radiation delivered via fallout and more sunlight reaching plants in the southern hemisphere. Both of these factors will mean the survival rate in the southern hemisphere will be greater than that of the northern hemisphere. In all likelihood, the much greater.