So, 1800ish weapons operational for each of the US and Russia?
Well, first of all despite being operational not all launch systems will be able to fire at a moments notice. For example, of each sides' SSBN fleets about one quarter to one third are usually in for deep maintenance and some proportion of the remainder will be down for lighter maintenance. So, of a nominal strength of 336 Trident SLBMs (assuming one each for all tubes on the 14 Ohio class still set up as SSBNs) in service the Yanks can
probably fire somewhere between 150-250, take the lower end if the war starts suddenly, or the upper end if steadily escalating tensions. Not sure of how things would go with ICBMs, but would expect some proportion to be unavailable and suspect the airforces of both sides probably loose 10-20% of nominal strength being non-operational at any particular instant.
So, of a nominal 1800ish active arsenal I suspect the number available for rapid launch is probably closer to 1300-1500ish.
Next, out of what's available to fire not all of 'em will successfully launch and not all of the warhead will go bang. Even during peacetime tests (usually associated with lots of TLC before the actual test) failure rates are significant: looking at
Minuteman II tests, there have been just over 300 test launches (both development and operational) with about 6 failures overall with three of the failures occurring in the last 60 tests... so, let's call it 5%. There's a fair prospect that without the TLC applied to selected test missiles the rate would be higher, let's arbitrarily go with 10%. On top of that, warheads may not work... No data on current weapons but back in the 1960s over 75% of the W47 warheads were believed to be unreliable. Again, let's suppose we have a 10% failure rate...
So, assuming all nukes are ICBM/SLBM mounted (not true but aircraft you also need to factor in air to air losses etc.), with 10% failure rates for launch and detonation and of your 1800 nukes only 1500 actually are available, 1300 actually launch and fly, 1100 warheads actually arrive and go bang. You can probably add in additional failure modes (goes out of control on reentry; goes bang but off target; etc. etc.).
Finally, with only 60ish percent of your nukes likely to arrive any targets you utterly need gone are going to need
at least two nukes thrown at them. This will, of cause, further cut down on the number of targets available... of cause, without knowing choice of targets and the counter-force vs counter-value balance we can only guess. Hardened targets (missile silos, command bunkers etc.) will likely require three or more to be 100% confident of landing bang close enough to guarantee a kill. Large cities will need several hits to actually wipe them out (assuming counter-value focus and no moral concerns over the implied war crime).
So, after assigning multiple nukes to targets that really need to be gone, of the 1100 warheads that go bang you've probably got anywhere from 300-800ish actual targets.
If you're Russia, now consider spreading those 300-800 targets over Europe, the US, US allies outside Europe and possibly other parties (China, much of South America, etc.) you want to make sure can't exploit the aftermath.
Long story short, the bang alone from the US or Russian arsenal will do relatively restricted damage and probably not really set civilization back at all. Possible flow on effects (cooling -whether a bad six months or full on nuclear winter- buggering up crops; implosion of the global economy; potential large scale migrations; etc. etc. etc.) on the other hand...