Very fast.
But it depended on the case if the tribes were conquered or not. For instance, Spaniards conquered about half the territories of today's Hispanic America, and natives lefts of theirs area of influence were not converted to christianity, obviously. In Brazil, almost half the territory was not conquered by the Portugueses, either. Those natives that were not conquered usually preserve theirs believe up to this day.
Now, Natives that got conquered usually passed by a process of religious syncretism. The Catholic priests, that were quite smart in preaching theirs religion, usually got Natives into the Church by addopting local religious ideas to Christianity.
For example, it is quite obvious that Mayans and Aztecs, that practised human sacrifices regularly, got fascinated with the shocking images of a bleeding Jesus in the cross, despicted with luxury of details. In certain sense, from their point of view, Catholic Christianism was just a variation of theirs own believes, and it is amazing the coincidences in both religious forms. The Virgin of Guadaloupe, for instance, seem to be for ancient Indias a syncretic form of adoration to an ancient local godness, transformed into the Catholic cannon. The same is appreciated quite clearly with other religious figures in most Latin America, where ancient gods more than replaced mutated to new ones. Quetzalcoatl got converted in Jesus, sort of speak.
Even today, the cult of the Saint Death, in Mexico, seem quite out of place with Catholicism, and I believe is rooted in an ancient mentality that has never dissapeared.
Ancient Amerindian traditions never died. Of course, human sacrifices were forbidden, but still today you can see Quechuas performing the sacrifice of the llama, in honour to the Pachamama.
In any case, Natives and today Latinos are sincere in theirs Catholic believes, and they know more about that religion that more Europeans.
In certain way, many Latinos consider that religion was one of the possitive things brough by Europeans to the Americas, and everyone knows that priest have usually being at the side of the Indian and the poors rather than supporting the injustices.
In any case, the changes were fast. In a century all the Natives and mixed people in the Spanish Empire of the Americas were Catholics. Under Spain rule there was not choice but being Catholic because that was the official religion of that state.
Pinguin