In the opening for Atlas Medieval America, the words include
"In most of the world, the New Middle Ages were merely a rerun of the Old Middle Ages. "
One question is...how does it go back to being a rerun? The reversion to medievalism is ostensibly organic; People aren't looking to LARP on purpose. So one has to rationalize the steps to everything repeating itself. Would we have the same nation states as the Middle Ages? In places the nationality more or less existed in the Middle Ages, I would say "yes". I also propose something like a hundred year "grace period". For instance, the unification of the Iberian peninsula is considered to be one of the things that ended the Middle Ages, but I think we can consider a relatively whole kingdom of Spain to be part of the New Middle Ages, largely because it has a current monarchy that might be a little tough to divide into sub-kingdoms. (Though chunks would probably be bitten off here and there. Maybe a modern day Grenada as a haven for European Muslims)
Much simpler to define are nations with present day monarchies and rough medieval equivalents would be two countries the White mentioned; England and Japan. They have established royal bloodlines, and a history of monarchies. There's also Denmark, Sweden, Norway, Monaco, and Morocco. One could also expand the scope of the Vatican. Most other monarchies (especially outside Europe) postdate the Middle Ages, so there's no "need" to reestablish them. The trickiest one is perhaps the Netherlands. As a kingdom, it's relatively modern, though it has been a de facto monarchy throughout its somewhat modern history. However, there was more or less a Netherlands ruled by a House of Orange in the 1500's, so we can maybe make an allowance for that. But it may also be swallowed up some new version of the Holy Roman Empire.
The Holy Roman Empire itself is some kind of mishmash misnomer, but with the Catholic Church returning to dominance in the politics of Europe, and Germany as an economic power that's falling but doesn't quite want to do the "conquering thing", one could see them kind of joining forces as something they insist is a "loose confederation", but most outsiders and mapmakers are like "Yeah, it's the Holy Roman Empire". Likewise, there's the Byzantine Empire, which would not actually reform, but any Orthodox practicing Baltic states forming into any larger entity would be called a new "Byzantine Empire" by observers, especially as they never really called themselves "Byzantines". (They thought of themselves as Romans) As to whether they would take back their capital city--perhaps, although it Istanbul could remain in Turkish hands and still fall under the "grace period" rule. But a defining aspect of the Middle Ages was that Constantinople was THE biggest city in Europe.
As for nations that more or less existed in the Middle Ages, and might revert to monarchies when they currently don't have any, like France, Hungary Poland--it might be hard to picture how France could establish a new royal family, since one would think there might be some lingering hard feelings about and very little monarchies. But France was THE monarchy of the Middle Ages. The best way to handwave is that France devolved into a bunch of feudal states, and the top strongman would marry into another royal family for alliance/power, which would mean France now has a reigning monarch. Largely the best way to establish kings in nations that had done away with the concept.
Many of the nation states in places like Africa and the Muslim world were established by colonial European powers, so it's very easy to see them washed away by the tides of war and new regimes. The major exceptions being the aforementioned Morocco, Mali, Ethiopia, and Egypt. Most geographic regions were established by ruling dynasties and tribes, and that would probably be the case now, but there would almost certainly be an concentrated attempt to wash off Colonialist history.
But now we come to the Americas. Now, "America" is sometimes used to refer the United States, sometimes both the North and South American continents. How much is the Middle Ages a "new experience" in America as a whole? Well, let's just say a good chunk of Brazil, the Amazon, and a few interior deserts have almost reverted completely to tribalism, if not wilderness. Very distant cities probably can't sustain themselves, quite possibly a third of South America's geographic area does in fact resemble the pre-Columbian era. Then you have the Andes and Mesoamerica. These fall into "the grace period" of the 1500's. We would not see the return of Incan or Aztec Empires, but something akin to a Latin overclass ruling a people with indigenous ancestry who are ostensibly Christian but speak their own tongues and practice their own local folk beliefs. This would would especially be true for the Andeans, who were able to weather the European cultures compared to other natives.
The major "new experience" would be for the Southern Cone. Much more like the U.S., in that they were colonized relatively late, and with a less indigenous-descended population. The land and number of people would be a mirror for Northern America, but with less than half the land area and population. In general, the medieval USA would be truly be unique in its scope.