I think the closest parallel is WW1. When the war started, no power was really prepared and had only months worth of vital provisions (if that) and very quickly things degenerated into a hand-to-mouth slugging match where what mattered was industrial power and access to world trade and where pretty much all of the rules were disregarded. (For example, poison gas, which everyone used and everyone had agreed to outlaw a few years before, abuse of PoWs, abuse of captured civilians, the free use of lies by all sides, like when the Entente promised territory to Romania if they joined the war while simultaneously agreeing among themselves to not honour those promises, the casual disregard of the sovereignty of neutral nations when it seemed some benefit might be gained from it - witness the invasion of then neutral Greece by both sides, the invasion of Albania by the Greeks, Serbians, Italians and later even Austro-Hungarians, the invasion of Persia by Russia and Britain, unrestricted submarine warfare in the Atlantic etc, etc.) In the end, using only conventional weapons, millions upon millions of human beings had been slaughtered, empires broken, and the entire political trajectory of the world had been sharply altered.
Similarly, no-one is prepared for a great power war in the modern world. Similarly, modern industry and communications would mean that even without nuclear weapons it would be thoroughly apocalyptic on a scale that would make WW2 look like a storm in a tea cup. I'd give a narrow advantage to the US and its allies in any war, simply because the US is best placed to command world trade in such a war, allowing them to do what the British did in WW1 and use the world's resources to feed their people and their war factories while similarly denying those resources to the Germans. Also, with the American advantage of decades of higher military spending than either Russia or China would give them an advantage in the early war that would enable them to claim advantageous positions just like the better resourced (pre war) German army was able to use its advantage to occupy much of the French industrial heartland and ensure that the bulk of the war was fought on foreign soil.
However, the industrial power of the US and China is so close now that the US would have a very real risk of losing such a war if they make mistakes (just as Britain faced the risk of losing WW1).
Russia, while in no position to win a war against the US, is still one of the great industrial powers of the world (though much declined from Soviet times) and a conventional war would likely take many years and be extremely costly and painful for the US/NATO.
It's hard to see Russia and China fighting together (given that they are not currently allied, don't currently show any desire to be military allies and intentionally joining such a war would be an act of national suicide, even if the Russo-Chinese alliance won), but if they did it would make for a significant challenge in conventional warfare.
While the US might fight Russia and/or China alone, no way is the rest of NATO going to fight without the US - the other NATO states just aren't equipped for a great power war without the lead member.
And of course, given how brutal such a war would be, it's basically ASB that any gentleman's agreement could keep nukes from being used, and there's basically no point in using nukes against another nuclear power unless you use them all (since using all of them might just stop the enemy from annihilating you with their nukes). Consider how many tactics and weapons were considered too vile to use at the start of WW1 and WW2 (like poison gas and firebombing) that were embraced energetically as the wars dragged on.
Great power war in the modern era basically means the extinction of the great powers who fight it.
fasquardon
Honestly, I'm pretty sure your industrial out put really doesn't matter in a WW3 scenario modern day weapons from what I heard seen and read are far far too complicated for your average steel mill or car factory to produce quickly sure you would see things like 24/7 shifts for major arms factories things like Boeings St Louis Plant or Lockheed Martin same for Dassault in France etc. etc. etc. to quote Calbear here (Jesus is he a reverse for quotes):
Because, as I suspect you already know, B-2s are closer to the construction of a top end watch than to the building of a Chevy. That level of precision takes time. It didn't matter in WW II if a rivet on a B-29 was 1/64 of an inch too high, either due to a supplier or assembler error, if the B-2 has that sort of error its stealth drops by 95% and a $2B aircraft and two exquisitely trained American pilots wind up a smoking hole in Outer Shithole.
In any war no nukes, I'd almost certainly hand it to the US and friends power projection large stockpiles of gear. It will be bloody and supremely costly i think bare minimum the US is gonna lose over 1400 aircraft in operations probably at least one probably two or three carriers and a fair number of SSNs