This may even happen in the southern hemisphere which may not be the ground for a nuclear war. We assume now that India and Pakistan will keep out of such a war. The negative temperature would continue for several months resulting in a short-time cooling. It can happen only in Hollywood films. Exploded gas fields would release very large amounts of unburnt methane, ethane and propane.
Weapons are bad, no matter what they are and how they look like.
In the hands of an irresponsible possessor even an ordinary AK47 rifle or the like can be deadly, not to talk of weapons of mass destruction. Let us not forget that many more people were killed during the second world war by conventional bombs than by nuclear bombs dropped on Hiroshima and Nagasaki. Let us hope that Prof. Hawking is right. They are particularly effective in the investigation and rational utilisation of water reserves below the Earth's surface, or groundwater.