For the most part the ruling Nahua nobility kept right on being the ruling nobility. Particularly in the first couple decades after the fall of Tenochtitlan, the Spanish had neither the numbers nor the inclination to displace the local ruling class. Lockhart, in his Nahuas After the Conquest actually calls the early colonial period a time of stasis that maintained earlier forms of governance and tribute. Later, of course, the increasing numbers of Spanish colonists and declining numbers of Indigenous peoples would undermine these older political forms and effect a Hispanization of society. This transformation did lead to the paring away of power from certain more powerful polities, as the informal power they held over subordinate cities and towns could be eroded through formalization of relations, but that is a little outside the scope of your question.

Indigenous elites, however, often just moved into re-titled official roles. So a tlatoani would still be a the ruler of an altepetl, only now his official role would be as a gobernador of a pueblo, replacing the "-tzin" honorific with "Don" and Doña". Direct descendants of Tenochtitlan's ruling family, for instance, continued to rule (San Juan de) Tenochtitlan for decades after the surrender of Cuauhtemoc to Cortes. Cuauhtemoc himself continued in his role as tlatoani after the surrender and even accompanied Cortes on further expeditions (including to Honduras where he was executed on charges of plotting to kill Cortes, charges now thought to be false). Several of our most important early histories of pre-Hispanic Mexico actually come from Nahua elites who were telling the story of their own ancestors and who served in various official capacities. For example, Chimalpahin, Fernando Ixtlilxochitl, and Fernando Tezozomoc, who were, respectively, descendants of the ruling families of Chalco, Texcoco, and Tenochtitlan.

The descendants of the ruling family of Tenochtitlan are the best documented, in part because Cortes had a child with a daughter of Motecuhzoma whom he legitmized and also because the family itself entered into the Spanish nobility as the Counts (later Dukes) of Moctezuma de Tultengo. To return to the daughter of Motecuhzoma though, Isabel Tecuichpo was accepted into the Spanish nobility herself and her daughter with Cortes, Leonor, would marry the Spaniard who would go on to found New Mexico. That particular branch of the family was actually instrumental in the colonization of Northern Mexico and the Southwest. Chipman's Moctezuma's children: Aztec royalty under Spanish rule, 1520-1700 does a good job of tracing the fortunes of this particular lineage.

The elites were not massacred or carried off into bondage. Many lesser nobility did find their positions undercut by the new Spanish political roles, the imposition of the casta system, and the devastation that epidemics in the mid and late 16th Century wrought upon the indigenous population, moving into the general population while often serving in more unofficial capacities or as part of the colonial bureaucracy. The upper nobility of the various altepetli, and of Tenochtitlan in particular, persisted in their roles or moved into roughly equivalent positions in the Spanish system, even entering the Spanish nobility.