The Russians would launch all their missiles and bombers at the US –Russia would launch all their arsenal of nukes against all major NATO cities. Russia would die, but then so would the United States.
It is called Mutual Assured Destruction (MAD) and its what has stopped nuclear countries from having a nuclear war.
It takes about twenty minutes for ICBMS to reach their target, and both sides carefully watch the other for any sign of a launch, and a massive launch is impossible to disguise. In the space of the twenty minutes, the leader of the country is notified, and authorizes a retaliation strike. The military prepares its missiles and bombers and assigns then targets and launches them BEFORE the enemy missiles arrive.
The country that launched first is the last to die, but die they will.
If US should launch 1000 nuclear weapons at Russia, Russia is capable to launch 1200 missiles at US, and each of their missiles can contain up to 36 warheads. Twenty minutes after they launch over 20,000 nuclear warheads ‘appear’ over the United States moving at 430,000 mph to 630,000 mph and they are only 110 miles from their targets.
Within two minutes (or less) all those warheads will explode over the US, and most, if not all of the country would be destroyed. By the time the Russian response hits US, Russia would have been obliterated, but they will take the US with them in death.
Both the US and Russia traditionally have a Counterforce doctrine, where their goal is to eliminate as much of the other country’s military and ability to retaliate as possible—especially in a preemptive strike.