Typically, Jonah is dated to the period of Jeroboam, i.e 780 - 750s BC, whereas Nahum is dated to either shortly before (615BC) or after (612BC) the fall of Assyria (and its capital, Nineveh). Thus, Nahum is at least 150 years after Jonah.
So, to answer the question directly, No - Nahum is not the promised vengeance of God against Nineveh promised by God in Jonah. If it were, then God is not "slow to anger, but abounding in compassion," as Jonah himself points out in chapter 4, nor is the "incomplete ending" of Jonah to any effect.
The key date that differentiates these works, however, is 722BC, when Assyria conquered the northern kingdom of Israel, leaving only the last two remaining tribes of Israel - Judah and Benjamin. Jonah presumes a full, diverse Israel, whereas Nahum is only addressing the remaining "Israel" in the sense that most people think of Israel today. The fuller complement of Israel would have been more steeped in idolatry and the "sins" people think of today.
When Jonah goes to Nineveh to preach repentance, the amazing thing is that Nineveh does repent, much to Jonah's annoyance. That generation is completely spared the wrath of God and lives, so to speak, happily ever after.
It isn't for another 150 years (think the time between now and the American Civil War) that Nahum is writing. He is fortelling the doom of Nineveh in order to encourage (Southern) Israel from prematurely capitulating to Assyria. The later parts of 2 Kings and 2 Chronicles tells the stories of Hezekiah and Josiah, set in this time period, and the constant question is whether Israel should rely on God, or trust in some bigger foreign power like Egypt or Assyria, the two big players in the region. Imagine, for example, you're Belgium just before WWII. Do you rely on God or make peace with either Germany or France, and you're beginning to get the picture.
As it happens, Egypt itself had fallen shortly before Nahum, and Assyria does fall very shortly after (or if you don't believe in prophecy, before) the writing, so in any case, its good advice.
Unfortunately for Judah, however, Babylon is the instrument of Assyria's departure from the scene, and within only about 40 more years (587 / 586 BC) they will invade Judah itself, thus destroying the last vestige of the kingdom, and sending all the peoples into exile.