If we want something like the above few maps, we need one, domesticated animals beyond dogs, turkeys, and muscovy ducks, and two, better staple crops than maize. If we have that--let's say we have domesticated reindeer (caribou), mountain goats, and ducks (mallards, like in the Old World)--and let's add a few Eastern Agricultural Complex crops (amaranth species) plus some innovations like wapato--then we can truly have Northern American civilisation. For South America, we merely need an earlier spread of quinoa and potatoes from the Andes to places like central/southern Chile and the La Plata basin. If we have early sailing (cotton sails in Mesoamerica/Andes, and sails from Indian hemp/dogbane in Northern America), we can have the trading and cultural influence which can really spur indigenous civilisations to develop. Then it's pretty much dependent on local societal development and the prominent figures it raises up to build solid states.
I don't think colonisation as a whole can be avoided without some really old and shaky PoD. Anyhow, let's assume this happens and the Europeans, Asians and north Africans reach the New World, Sub-Saharan Africa and Australia/Oceania around the same date as IOTL with all powers being the same.
Here's my take at North America:
There were a myriad of undeveloped tribes in North America around 1492 with the most advanced being those of the Mississipi river and cities such as Cahokia. These at the time were collapsing due to food needs (they mainly depended on corn, which depletes the soil quickly and limits the amount of time a sedentary culture can remain in a place, which limits the potential to fund cities). European explorers reach the east coast of the continent and trade ocasionally with relatively advanced Indians such as the Powhatans or the states that De Soto found in the southern Appalachians. These states would be destroyed by European or African diseases, probably killing 80% of the population or something on the lines. Plague would extend across the continent (and the whole Americas) over time, like a wave of death flowing east-west. New crops and livestock would make it from Europe around the 1600's with Indians likely having enough time to partially recover from the pandemic.
The 16th century was apocalyptic for the Mississippians, their cultural relatives, and other eastern Indians. Lengthy drought decimated them, followed up by European-introduced epidemics. De Soto wandered through and contributed in the destruction of a collapsing civilisation. Cahokia had collapsed before then, with the center of power shifting to the (now destroyed) mounds in East St. Louis/St. Louis around the 14th century. But like a lot of Mississippian settlements, they collapsed by the 16th century and became the "Vacant Quarter", with their indigenous people migrating around to form the nucleus of new ethnic groups. You'd need something to prevent that and keep the latest Mississippians (like those who lived at the 15th century Kellytown site near Nashville, TN) around. This would imply an earlier PoD, likely implying more developed crops and more domesticated animals (the Mississippians don't seem to have domesticated either turkeys or mallards, despite both species being found in the area).
Assuming advanced civilisation continues in Mexico with trade-based people such as the Huastecs, we may see trade routes being stablished between Mexico and the Mississipi, with colonies and outposts being founded in the northwestern part of the gulf. Eventually the Mississipi would become the heart of America, likely hosting a great empire comparable to Rome in being the base for North American culture and laws (but likely being far smaller). The southern US would develop faster with several states popping up across the Appalachians and the east coast, likely fighting amongst themselves for access to trade routes and resources. The northeast would develop slowler at the beginning but would grow exponentially faster, being closer to Europe and having a similar climate. I see one or several European-led states seize large chunks of New England, Canada and the Atlantic Coast to later collapse and give birth to other independent states such as an enlarged Iroquois or an Huron-Canada analogue. Further in the northeast, I see the fisheries of Newfoundland too attractive for the Europeans to simply renounce to them, and even if I'm trying not to, an European state would probably conquer or seize control of Newfoundland and nearby lands in the Gulf of Saint-Lawrence, with mestizo states surrounding it. Forgot to mention Florida, but the area would more likely be closer to the Caribbean sphere than the Mississippian one, with maybe a syncretic culture.
You need sailing for that, and it's fortunate that cotton sails will facilitate things in that regards. Earlier sailing would help big time in moving cultural innovations from Mesoamerica to the Caribbean to the Mississippians. But considering the demographic law of exponential population growth once certain factors are removed (for the Mississippians, perhaps we'd have metalworking, more plant/animal domestication, etc.) it's likely the Mississippians fully emerge as a civilisation. Something likes Monks Mound at Cahokia would be commonplace at a major Mississippian site. I think Mississippians would fight amongst each other for dominance, and considering local resources, it's likely a site in the Nashville area (near Nashville itself, the many important mounds were destroyed very early, but the Nashville Basin and surrounding areas has examples like Castalian Springs, Mound Bottom, and other local sites which were important Mississippian centers) can build a solid empire to the coast like in the Old World. Muscle Shoals in Alabama is even better, since it's on a natural chokepoint of the Tennessee River and forms a nice point to mediate trade with the Gulf. Even OTL, competition in this area occurred, hence why Mound Bottom along the Harpeth River in the Highland Rim declined OTL before sites in the Nashville Basin. The Nashville Basin has rich agriculture and the rivers to transport things, so can serve as a perfect center of civilisation. The Highland Rim and Cumberland Plateau, where temperatures are a few degrees cooler in the humid summer, can potentially support any far southern extension of reindeer or mountain goats (reindeer do still have the issue of parasitic worms from local deer). I'd love a TL where the famous "Tennessee fainting goat" is actually a breed of mountain goat, which is normally powering local indigenous civilisation. A Tennessee River civilisation would have a lot of saltpeter (easy transition to gunpowder weapons) but also a lot of natural iron and copper.
Reindeer in New England would mean highly different circumstances for the cultures there. The northeast lacks the cotton which makes the sails, but I can't find a reason why Indian hemp (
Apocynum cannabinum) can't fill that role. Give the northeast woodlands Indian hemp and some good ship designs, and they can exploit the Grand Banks and fight the Basque and other medieval Europeans for the rights to fish there.
The central plains and the Midwest would develop slowly, with Mississipian or Iroquois/Huron outposts on the Great Lakes and the Mississipi and Missouri rivers. Plain tribes would be influenced by these, with some resorting to agriculture while groups further inland would remain hunter-gatherers or may switch completely to bison hunting. If you add horses and other Eurasian steppe domestic animals we could see a confederation of aggressive and expansionist tribes always wanting to seize more land for their horses and bison-hunting. The Mississipian Empire would be constantly harassed by these tribes in a Rome-Germania analogue. That area would be civilised over time, but at a slow pace and keeping most of it's traditions, comparable to Mongolia and China.
OTL Plains Indians were influenced by Mississippian culture, and if successful farming can extend far enough inland and far enough north to support a sedentary year round population (at the height of the Medieval Warm Period, the ancestors of the farming/sedentary Plains Indians i.e. the Mandan had this going for them), they can do pretty well. The Great Lakes will be a natural center of civilisation, especially since there's plenty of copper-and iron-around.
The Rio Grande area would be influenced 2/3 by mexicans and 1/3 by Mississipians, however due to the sheer distance from both states it would create it's own distinct culture, specially when referring to OTL New Mexico. I see the area south of the Arkansas and east of the Colorado become it's own center of power over time, possibly displacing the Mississipians or the Mexicans for influence in neighbouring states in Sonora or Kansas. California has potential to become it's own nuclei, however it would appear later and most of the area would already be influenced by either the Rio Grandese or Mexicans. I could see different tribes in California resort to different ways of life, such as a trade-based Chumash state or a Californian-valley with agriculture as it's main export. Over time that central valley state would grow and expand to encompass most of IOTL California, but would be locked by the different climate in Oregon and the Rocky Mountains to the west. I have no clues over who or what could control the Great Basin and the Rockies, with the area being sparsely populated and mainly barren. Some strong and small city-states could appear in river valleys, but most of the area would be controlled only in theory, maybe with horse, llama or other big animal caravans going across the basin. ´
California in the best case scenario would be maritime, with likely the Chumash dominating and the people of the Central Valley adapting. Key sites like the Bay Area would be conquered by the Chumash. The people of the valley would be a divided population raided for slaves until they consolidated into local states. The Great Basin would perhaps be comparable to the Bedouins. They'd have mountain goat pastoralism and enough farming in the fertile river valleys to thrive, and also a key niche to move goods between various regions.
Oregon and the Columbia river would use well as an agriculture and fishing area. I can see a trade-based empire being formed with it's nuclei around the Columbia river, possibly having colonial outposts across Cascadia. I guess it would have a rivalry with California, having a volatile border as both sides would want control of the Jefferson State area. I guess tribes in southern Alaska and northern British Columbia would form some sort of confederation to stop the Cascadians (or Oregonese or whatever) from conquering the area. I guess they would succeed. And finally we have the far north. Athabasca and the area dubbed as "American Siberia" would either be it's own state (American Russia, but far weaker) or be settled by the Plain Tribes or the Inuit, with maybe some Cascadian settlements on the other side of the Rockies. Then we have the Inuit, which would form a contiguous ethnic group across the Arctic shore of the continent. It could either remain unified or split into different trade states, but it's overall population would be abismally low. Finally we have the Creek and other northwestern Canadian tribes, which would be weak in a land without agriculture potential and would likely depend on the Europeans or the Hurons for food and other things, probably would be the weak pushover not worth conquering.
They'd be most influenced by the Puebloans, but it is curious how marginal OTL Indians along the Rio Grande were.
Subarctic/Arctic America could never form its own unit without a very different Inuit/Athabaskan native population. As for the "state of Jefferson", OTL provides a nice border--the Klamath/Modocs of Southern Oregon raided California peoples like the Shasta and Achomawi for slaves. The furthest north counties in California OTL provides a nice border for this Indian culture.