As I see it, the biggest risk of nuclear war under Trump is either an overreaction to an attack against the United States (1), or enemy states taking advantage of an American withdrawal from protection of its allies (2).
For example: (1) There is a major terrorist attack on US soil, killing thousands of people. The public anger, fanned by the anti-Islamic rhetoric of the Trump campaign, is demanding a military response. Some evidence points to a link between some Middle Eastern rogue state and the attack. Can Trump resist the popular anger and stupidity? He hasn't shown these traits in the past.
Or (2) Trump indicates a cooling of support for South Korea, pulling out some of the US troops stationed there after South Korea don't 'pay up' as demanded by Trump in the campaign. With the militarised border always on the brink of war and North Korea emboldened by Trump's actions, some miscalculation/accident/aggression triggers a heavy artillery assault on Seoul from the North. Trump responds with a nuke, risking war with China.
And a million other possibilities where Trump's rash temperament, lack of clear thought and short-sightedness result in a mushroom cloud.