Trump is a manipulator and he's completely bluffing about the nuclear arms race to distract people from the real issues like his conflicts of interest. Besides, he still can't fathom that most celebrities do not want to attend his inauguration and threatening a nuclear arms race is his knee-jerk reaction to not getting his way.
Despite Trump's astonishingly ignorant remarks about nuclear weapons before and after his election it is very unlikely that a nuclear war is around the corner. The principle of mutually assured destruction still holds true in the post-Cold War era and no leader, not even Kim Jong-Un, is foolhardy enough to risk the total annihilation of their own citizens.
Hopefully, any reasonably intelligent person would realize that the start of a nuclear war would be the end of mankind. Is there really such a thing as a limited nuclear war? This may be why one has never been initiated before now. We have had the means for a long time.
I hope not. A nueclear holocost could be the result of a nuclear arms race, but much like how the nuclear arms race in the 1960-80's did not result in a nuclear holocost, this arms race (if it happens) probably won't either, because nobody really wants that to happen and we want to live.