One nuke wouldn't blot out the sun.
I don't know precisely how it was done in the books, but this is a suitcase nuke. The general theory is that if a hundred large cities were burned, all at once (wouldn't matter if you used nukes or a lot of conventional bombs), that might trigger a temporary nuclear winter. Not The Night Eternal.
I suppose you can say it's one of those special bombs, that turn up in stories with depressing endings (Beneath the Planet of the Apes comes to mind). But there was no indication of that in the story, and it didn't even go off in the city. Same exact thing happened Rise of the Dark Knight, and everybody was fine. Yes, I know this doesn't prove anything, and that movie didn't make a lot of sense either.
I've been very patient with things like this with regard to this series, and of course I'll watch next season. The finale was, I think, the weakest of the three so far, but still many good moments. Maybe they should have just dropped the Night Eternal angle entirely. The Master had basically already won, at least as far as New York is concerned.
They are going to have a lot of built-in problems to solve next season.