>How would you go about invading Russia, as a senior NATO commander?
NATO would have to nuke Russia, then invade. Tank crew are protected from fallout radiation. If they have filtered air, I think they can enter "fallout plumes" right away. Soldiers not protected by tanks will be able to enter in about 3 weeks: weapons fallout is very different in character from the contamination from, e.g., Chernobyl or Fukushima. It dissipates much more rapidly. In fact, since the fallout plumes will cover only about half of the land area or less, the tanks can map out the locations of the plumes, after which the infantry might be able to enter the parts of the country missed by the fallout plumes well before 3 weeks after the end of the nuclear attack. (The fallout stays in one place after it has fallen out of the sky and has hit the ground -- or more precisely the fallout that does end up being blown around by the wind after it has hit the ground is small enough in particle size to not be deadly, though it will mess up your mucus membranes via beta radiation, hence my words above about filtering the air for the tanks.)
When Jens Stoltenberg says that a nuclear war cannot be won and must never be fought, his "cannot be won" is not literally true. He is saying it to emphasize that NATO would never even consider starting a nuclear war. And in fact I don't think the US or NATO ever would choose to start an intercontinental nuclear war, but it is very hard for the Kremlin to come to understand the US well enough to be as confident of that as I am (having lived in the US for over 60 years). Also, the people who run Russia and who will run Russia after Putin is dead are professional spies. They are evaluated by how seriously they take national security. Also, Russia has been invaded about 50 times in its recorded history: by the French, the Germans, a Polish-Lithuanian confederation, Sweden, the Turks many times, various groups (other than the Turks) looking to get slaves, Central Asian peoples and many kinds of steppe nomads (mostly notably the Mongols, Tatars and Cossacks). The whole country takes national security very seriously.
Of course NATO would want to evacuate its cities before it begins its attack. If it does, more than half of its population will survive the inevitable Russian response -- probably much more than half. (It's been a while since I saw the relevant papers.) Also, the US has spent many tens of billions on research into missile defense, and Russia cannot know with any certainly whether that research has born enough fruit so that the US can shoot down most of Russia ICBMs in a big war. Also, the Kremlin has expressed concern that the US Aegis system can shoot down Russian ICBMs, and now that Ukraine is good buddies with NATO, Russia has to consider the possibility of NATO's stationing many Aegis systems in Eastern Ukraine in addition to the Aegis systems already on US destroyers in the Baltic Sea.
In 1951 or so, China sent an army of about a million men against a large number of soldiers of the US Army. This Chinese army had the usual instructions from its political masters, namely, to kill as many US soldiers as possible and to destroy their equipment. They did this even though they would only get their first nuke in the 1960s whereas in 1951 the US had hundreds of nukes. Although the events I just described are a far cry from China's invading the US homeland, it does go against the notion that nukes are somehow a magical shield against conventional military attacks if even a non-nuclear military will contemplate attacking a nuclear power.
By the way, consider the motive of Beijing in 1951: the reason they risked getting nuked was to avoid having a regime (namely, the regime in Seoul) friendly to the US on their border. They preferred having a buffer state, namely, North Korea between them and any country friendly enough with the US to maybe agree to host US troops. They preferred it so much that they sent a million men and risked getting nuked. That is one of the data points that led Mearsheimer, Kissinger, Merkel, Sarkozy and many other security experts to criticize the plan of adding Ukraine to NATO. (Merkel and Sarkozy stopped their criticism because Paris and Berlin depend on Washington to guarantee their security, which gives Washington the last word on Paris and Berlin's security policy, so they went along with the plan even though that thought they still thought it was dumb.)