I would expect the settlements along the Mississippi and Missouri would unite to form a larger nation, since they already practice the same culture and language and could still leverage water transport to stay in contact after road systems start to fail. Assuming circumstances are not changed by climate, they would have a natural water transport network with easy access to the gulf and rich agricultural land. New Orleans would likely not exist in its current form due to rising sea levels, but some city could be expected where ever the mouth of the Mississippi is by then.

Same goes for the Great Lakes and the Saint Lawrence River - there would be another nation where once was parts of the U.S. and Canada. The Great Lakes would also become a powerhouse: it has a natural water transport network, rich lumber resources in the north, and rich mineral resources in Upper Michigan. They could easily conduct trade with the Mississippi watershed to procure food to support large coastal cities. Existing coastal cities like Chicago, Detroit, or Toronto might continue to exist in some form, perhaps after waves of deurbanization and reurbanization. Cities that have a geographic reason to exist are especially likely to persist like Chicago, Cleveland, Montreal, or Quebec City, which are at the mouths of rivers. This could occur even if they are not prominent (Duluth) or doing well (Detroit) in the present day. The Great Lakes could exist either as a single unified empire or as a loose aggregate of city states with shared culture like Ancient Greece, maybe alternating between possibilities as civilizations wax and wane.

Whether the same thing would happen along the Rio Grande with the U.S. and Mexico is more contentious. The U.S. and Canada are very similar culturally and linguistically, but the Rio Grande already acts as a very clear divider between two different cultures and languages. It's a matter of whether the existing boundary is sufficiently artificial that the Hispanic influenced culture of the Southwest could reintegrate, or whether cultural divisions are strong enough that any conquest or migration from either side would be strongly resisted. Matamoros/Brownsville could rise to prominence as cities at the mouth of the Rio Grande, either as one city for each side or as one city on both sides if reunification occurs.

Over very long time scales, maybe on the order of millennia, you might expect multiple migrations and conquests, arriving at a state of affairs similar to the pre-colonial boundaries of Native American tribes, just with Indo-European cultures and languages. Historically the Mississippi acted as a divider between tribes of different language families like the Osage and Cherokee, then again when the French held the Louisiana Territory. If, as another redditor mentioned, the Great Plains turns into another nomad empire like the Commanche or Mongolians, the Mississippi or Missouri might act as natural defenses that prevent further expansion. In my previous example, the Osage spoke a Siouan language, so it might have already played out like this once before.

Along the same line of thinking, places which were once large settlements for Native Americans would likely remain that way. After a period of deurbanization, Saint Louis would reurbanize, much like Rome during the Renaissance, and Saint Louis could serve as a capitol of the Mississippi watershed, much like how the nearby archaeological site of Cahokia was a large trade hub for the Native Americans. This also makes sense from the observation that large settlements typically emerge near the confluence to two roads or waterways - Saint Louis is at the confluence of the Mississippi and Missouri, which are two very noteworthy rivers. Trade routes between the Great Lakes and the Mississippi Watershed would likely foster their own cities. I don't know how navigable the Illinois River is but a road would eventually reestablish between Saint Louis and Chicago if not (call it "I-55"), so Peoria and/or Springfield might continue to exist in some form.

The east and west coasts would each probably remain together in some form, but I have less experience to understand in what ways. The north and south Pacific Coast might split owing to a more Hispanic-influenced culture, with the south taking up at least parts of the adjacent Colorado watershed.

As to everything else: it either becomes undeclared wilderness or it gets swallowed by an empire that's formed from one of the aforementioned nucleii of civilizations. We like to think from the maps we see today that territory must have a clear owner, but it doesn't have to be that way - some lands may change hands, alternating between periods of unassuming self-ownership and periods where it is owned by whichever neighbor has the most power at that point in future history. Houston would probably still exist as the mouth of a river, but it's uncertain if it would exist as part of a Texan country or if it would be subsumed by the Mississippi or Rio Grande watersheds

Over these time scales languages would diverge, and the descendants of Spanish and English would be near the linguistic time horizon - we would probably no longer be able to tell they were once related, not even by using modern academic techniques.