The history of the state later known as Mexico began when in 1250 the people of the Tenochca migrated to the valley of Mexico. In 1299, Cocoxtli, ruler of Culhuacan, allowed the Mexica / Aztecs to settle in Tizapan. 1325, they founded their capital of Tenochtitlan.
In 1376, the Mexica elected their first tlatoani (=great speaker; means emperor), Acamapichtli. They were a tributary of Azcapotzalco at this time.
1481, the great Tlacaelel, cihuacoatl ("Prime Minister") of the Tenochca, died.
In 1502, Motehcuhzoma II became ruler of the Tenochca. The year when their god Quetzalcoatl was supposed to return passed without any European contact, fortunately.
1520, a small Castillian expedition (a few dozen men) met the Tenochca. They were defeated, noone of them returned. However, the Castillians beat the Tenochca at Tapachula (south of OTL Mexico).
History took an interesting turn in 1521 when Prince Alasdair / Alexander, the only surviving heir of the Norwegian (and Scottish) throne, went to Atlantis, where he spent some time around the Caribbean. He also approached the Tenochca and learned about their gold riches. In 1522, he came to the court of France, warned the king that the Quadruple Monarchy was about to conquer the rich lands of the Tenochca. He pointed out: "If the king owns their gold, he can afford to hire all the mercenaries in the world! You have to stop him now!" King Charles VIII made the right conclusion and started the Great Occidental War.
Meanwhile, in 1523 the Castillians wanted to send an expedition to Tenochtitlan, but the ship from Cuba was destroyed in a hurricane, and until replacements were found, necessary time passed. 1524, France managed to smuggle some hundred soldiers and advisors to Atlantis to help Cuitláuac, the military leader of the Mexica, as the Tenochca were now called in Europe. The Tenochca had to convert to Christianity, however. In the Battle of Cempoala 1526, the French-Tenochca defeated the Castillians.
Everything became much more troubled in 1529 when Smallpox hit the Mexica... however, the Allies of the Quadruple Monarchy weren't better off, so it was a two-sided damage. Motehcuhzoma II died, too, of the disease, Cuitláuac became new ruler of the Mexica.
In 1535, the Castillians drove the French-Tenochca back to the heart of the old Tenochca Empire, allied with their old enemy Tlaxcala. Only the lack of support prevented them from winning completely.
After the end of the Great Occidental War 1547, Europeans started regular trade with Mexica and Inca. They kept their independence because neither power wanted the other one to own their rich lands. Mexico adapted a pidgin version of French for communication with the Europeans, and the Inca lands followed suit.
Since ~1550, Cuitláuac started reconquering lost lands in Northern Mexico, extending to the Atlantic and Pacific again.
1552, Silver was discovered in Zacatecas. The Castillians went up the Rio Grande, founded settlements along it, to put a wedge between the French in the Mississippi valley and the Mexica. Later, those colonies would become Texas.
1554, a second plague hit Central Atlantis, and a third plague happened 1586. When the dust had settled, only 6 of 15 million Mesoatlanteans survived (and that's still better than OTL!)
In 1597, Axayacatl II of Mexico started to expand the Mexica Empire after long time again. After many years of war, the Chichimecs (N Mexico) were completely conquered in 1606; their lands were divided between Castillians and Mexica.
1619, Ahuitzotl II started building a Mexican war fleet, to protect their trade against pirates.
In 1633, the University of Tenochtitlan was founded. European knowledge would flow to Mexico, and they also returned something - Mexican influences entered European art, and in 1645, in the castle of Fontainebleau, the first "Mexican Garden" was made. (Mexican gardens are influenced by the Aztec way of gardening / planting crops, putting many different colorful plants next to each other, forming geometrical patterns, with colors matching of course.)
As a French ally, Mexico was dragged into the anti-French War. Things wouldn't turn out to their advantage. After the Battle of Metztitlan in March 1688, the Spaniards thrust deep into Mexico, where they linked up with anti-government groups. In the peace of Amsterdam, all of Mexico became a Spanish colony. The last Tenochca ruler Acamapichtli II has to go to exile in France.
Under Spanish rule
The Mexican gold had similar effects as in OTL, only later and not that extremely: Many Spaniards got very rich, but the country also started to suffer under inflation with time.
During 1730-37, uprisings happened in Mexico under a leader who claimed to be the rightful heir of the old native rulers and called himself Cuáutemoc II.
After the third French Republican War, Mexico became more independent, although the Spanish upper class still held all positions of power, and adhered to the Monteleón dynasty which had fled to Morocco.
1845/46, Mexico was conquered by New Rome.