Tawantinsuyu (also known as the Inca empire) came into existence around 1200 when its capital Cuzco was founded. Undisturbed by the events in the Old World, its history stayed the same as in OTL even centuries after the POD.
In 1493, Huayna Capac I became the Inca (ruler) of Tawantinsuyu. He didn't die in 1527, however, since there were no European diseases. One year later, in 1528, his son Huascar Capac was caught conspiring against his brother, the heir of Huayna Capac, named Ninan Cuyochi. Huayna Capac died in 1532, Ninan Cuyochi became new Inca.
During the years 1541-50, the Inca conquered the Chibcha lands in OTL northern Columbia.
After the end of the Great Occidental War in 1547, Europeans started regular trade with Mexico and the Inca empire. They kept their independence because neither power wanted the other one to own their rich lands. In 1549, French monks arrived in the Inca Empire to convert the population to Christianity. Since ~1550, after Mexico, the Inca lands adapted a pidgin version of French for communication with the Europeans.
Also in 1550, mining experts from Europe taught the Inca new techniques to get more silver from OTL Potosi. The Incas used their new riches to buy weapons and tools, and later books and machines from Europeans.
Plagues hit the Inca empire in 1553, killing a great percentage of the population. The Chibcha in the North and several more people (like the Aymara) in the South became practically independent again.
In 1570, French trading with the Inca brought the potato from the Inca empire to Europe. At first, it was treated as a curiosity, or as medicine.
With the help of his French advisors, Inca ruler Sayri Tupaq managed to stabilize his rule in the center of the old empire in 1571. Still in 1583, a second big plague happened in the reduced Inca empire.
1595, the Chachapoyas in the forests of Peru paid allegiance to the Inca again.
In 1614, the University of Cuzco was founded. The Inca started to acquire European knowledge, which was especially important for the mining.
During 1620-24, OTL Northern Peru and Ecuador were reconquered by Tawantinsuyu.
In 1655, the Border between the Spanish colonies in south Atlantis and the Inca empire in OTL Columbia was ruled by an official treaty.
During the years 1677-82, Inca defeated Aymara, reconquered the former South of their empire, now stretching to OTL Chile / Argentine again.
An Uprising of the Aymara 1707-10 was defeated, and as a consequence, the Aymara were displaced to the Southern tip of Atlantis.
In 1718/19 however, after their ally France had been defeated in the anti-French War, the Inca lost a border war against Spain. They had to pay a big tribute in silver (which was more important for the Spaniards than gaining land).
1723, the Inca made the first trade agreement with the Russians.
Since France became republican, Tawantinsuyu saw one land after another fall to the growing empire of New Rome. Still, they didn't care. The end came during 1826-29: Lacking money, emperor Alessandro I attacked and conquered the Inca empire. Their gold and silver mines now worked for his New Roman Empire. The southern tip of their empire became the independent state of Aymaria.