I have an idea for the Americas, but not yet for elsewhere.
Had the medieval warm period been warmer and lasted longer, the Scandinavian Vikings may have managed to establish small, permanent colonies (cf Vinland) along the coast of the western North Atlantic, from Baffin Island to Newfoundland.
Once they ventured further south than Vinland, they would have eventually found some very temperate climes in places like Long Island, the Chesapeake and the Carolina tidewater. Given such fertile lands, they would surely have developed trade and perhaps scattered colonies, which would slowly creep southward along the coast, only stopping when they reach Florida and tropical diseases. The economic and political chaos in Europe precludes mass colonisation, but maintaining contact is a real possibility.
This would have allowed the Columbian Exchange (of diseases, animals and technologies) to occur both much earlier, and much more slowly. The slowness is important, because it gives the natives of North America a few centuries to adapt to European introductions. It is likely it would have diffused completely (except maybe some of the more remote Inuit and Amazonian tribes) throughout the Americas by say, 1200. This would lead to a transition from hunting gathering and nomadic farming to sedentary farming and trade specialised cultures.
The rise of loose nations would divide eastern North America among the Iroquois Nation in what is now the US northeast, the Mississippi Nation (including roughly MO, TN, KY, LA, AR, MS, AL), a Plains Confederacy of some sort, a Great Lakes and Laurentian based fresh water maritime nation using Viking sailing tech, and an assortment of minor states, such as perhaps buffer states between Mississippi and Iroquois in Appalachia, Georgia and the Carolinas. The original viking colonies would likely have independent status as enclaves, mostly along the coast, but also perhaps with settlements in spots along the river valleys, such as the St. Lawrence, Hudson, Delaware, Chesapeake and on the VA tidewater. Greater hurricane activity due to warmer weather would make permanent colonies further south along the coast too chancey.
One or more of these native nations would likely use viking tech to conquer, colonise or absorb the natives of the Caribbean.
This way, when European colonial powers arrive on their shores driven westward over the Atlantic by the sudden end of the warm period in say, 1500, they are met by a large number of natives bearing resistance to European diseases, carrying European weapons and riding horses, rather than the scattered survivors of massive plague they did find. The Europeans would also find natives more similar to themselves, perhaps even speaking European languages and having been partially Christianised. Had the Aztecs had these advantages, it seems unlikely the Spanish could have conquered them.
Any colonisation would have been more similar to that in the Near East or India than IOTL the Americas.