I'm imagining some sort of three-pronged strategy. One, espionage to convince people in the nuclear chain of command to disregard any orders to fire nukes. This would involve converting people that have likely been thoroughly vetted by the Russian government. It would also be risky in that all it would take is for one person to snitch for the Russian government to catch on.

Two, a cyber attack that disarms nuclear weapons firing systems. This would likely involve gaining physical access to many launch systems, infecting their computer systems, then letting the infection stay dormant without getting caught yet somehow activating it when necessary. Say for example they run a dummy drill without nukes, the infection could be discovered.

And three, a interception system for nukes that are launched. This would be the most risky because it would involve intercepting nukes immediately after being fired. For ICBMs, we'd have to get them right after launch since once they're in space, it's nearly impossible to intercept, especially after the warheads separate from the rocket. Submarine-launched weapons might be easier to intercept if they're strapped to a rocket until detonation. Bombs would be nearly impossible, but it would be a lot easier to intercept the planes they're on.

Overall, I would guess we'd be able to stop some Russian nukes from hitting NATO targets, but not all of them. It would be a wild guess to calculate the percentage that get intercepted/through. Russia has about 1,710 nuclear weapons deployed. Let's say they fire half of them as a retaliatory strike saving the other half as defense in case the retaliation stops a NATO attack. If only 1% of that half make it through, that's still ~~85~~ 8.5 nuclear strikes. ~~If only a 10th of that were aimed at major cities, that would be 8 major NATO cities that are obliterated and then require major recovery efforts.~~

Not one country is prepared to recover from a nuclear strike because that's virtually all natural disasters in one. Imagine the devastation if London, Paris, Berlin, Rome, New York, San Francisco, Washington DC, and Los Angeles. There would not only be major loss, but the rest would have to dedicate immense resources to helping those areas recover, further pulling resources away from defense and counterattack. ~~We would also have to consider that the other 75 nukes attacked infrastructure and military targets, so we'd be severely incapacitated.~~

tl;dr: stopping and surviving a Russian nuclear attack is practically impossible