The following is specifically centered on the "time of the end” (Daniel 11:40 through Daniel 12:9). But some excellent information concerning the changing of the political identities of these two kings prior to the 20th century can be found in the Daniel's Prophecy book published by the Watchtower and Bible Tract Society.
Concerning the "Time of the End”
Concerning the "time of the end”, Daniel 11:40a reads:
“In the time of the end the king of the south will engage with him [the King of the North] in a pushing.”
If “the time of the end” here means the same as it does at Daniel 12:4, 9, we should look for the fulfillment of these words throughout the last days. What immediately comes to mind when a "pushing" is described between two major powers during the course of our past century? What two "kings" have targeted fearsome nuclear weapons on each other and have engaged in high-tech espionage as well as diplomatic and even limited military offensives? This kind of "pushing" has even adopted a name: The "Cold War".
“Against him the king of the north will storm with chariots and with horsemen and with many ships; and he will certainly enter into the lands and flood over and pass through.”
The identity of the King of the North in the time of the end seems apparent when considering that the history of the past century has well recorded the King of the North's expansionism. But, even though—from the viewpoint of his rival—the King of the North has loomed as a menacing presence, he has not achieved world conquest.
Events To Come
But what happens next? The following is a brief excerpt from the 11/1/93 Watchtower, "The Final Victory of Michael, the Great Prince", pars. 10-14:
"Does the rivalry between the two kings continue indefinitely? No. The angel told Daniel: “There will be reports that will disturb him [the king of the north], out of the sunrising and out of the north, and he will certainly go forth in a great rage in order to annihilate and to devote many to destruction. And he will plant his palatial tents between the grand sea and the holy mountain of Decoration; and he will have to come all the way to his end, and there will be no helper for him.”—Daniel 11:44, 45.
"These events are yet future, so we cannot say in detail how the prophecy will be fulfilled. Recently, the political situation regarding the two kings has changed. The bitter rivalry between the United States and Eastern European countries has cooled. Further, the Soviet Union was disbanded in 1991 and no longer exists.—See the March 1, 1992, issue of The Watchtower, pages 4, 5.
"So who is the king of the north now? Is he to be identified with one of the countries that were part of the old Soviet Union? Or is he changing identity completely, as he has a number of times before? We cannot say. Who will be the king of the north when Daniel 11:44, 45 is fulfilled? Will the rivalry between the two kings flare up again? And what of the huge nuclear stockpiles that still exist in a number of lands? Only time will provide the answers to these questions.
"First, though, notice that these acts of the king of the north are not said to be against the king of the south. He does not come to his end at the hands of his great rival. Similarly, the king of the south is not destroyed by the king of the north. The southern king (represented in other prophecies as the final horn to appear on a wild beast) is destroyed “without [human] hand” by God’s Kingdom. (Daniel 7:26; 8:25) In fact, all earthly kings are finally destroyed by God’s Kingdom at the battle of Armageddon, and this evidently is what happens to the king of the north. (Daniel 2:44; 12:1; Revelation 16:14, 16) Daniel 11:44, 45 describes events leading up to that final battle. No wonder “there will be no helper” when the king of the north meets his end!"
For much more, see:
Kings of North and South - Links to Information (INDEX; Watchtower Online Library)
North Versus South, Michael Stands Up (si pp. 138-142; Bible Book Number 27—Daniel; Watchtower Online Library)
Time of the End (Insight-2 pp. 1103-1104; Watchtower Online Library)