When did Mesoamericans stop using their native weapons?

You falsely assume that indigenous weapons made from obsidian, wood and bone were effective against Spanish firearms, cold weapons made predominately of iron and usual Spanish armor. They were not. If they were Aztec empire would not be demolished so easily by relatively few Spaniards. That does not mean they were completely useless, especially if used en masse, but realistically speaking these were the weapons from Stone Age going against weapons from late Middle Age and early Age of Sail .

Back to Mexican War of Independence, rebels were certainly poorly armed, but this means they lacked modern weapons like muskets and cannons. They didn't lack cold weapons like makeshift spears, pikes, axes and other converted agricultural tools. These were of course made of iron and much better then weapons form Pre-Columbian era, but still not enough to defeat Spaniards. Rebels were of course aware of that, and tried at every opportunity to get hold of firearms, gunpowder and ammunition.

As for your main question, with Spanish occupation and subsequent colonization, Stone Age effectively ended in Latin America. Use of iron become widespread, both for tools and for weapons, and old craft of making obsidian and wooden weapons fell into obscurity because it was not needed anymore .

EDIT: At the request of commentators, some explanation. Obsidian is very sharp, sharper then iron (or steel) . Unfortunately it is also very brittle and breaks easily. You cannot fashion swords from obsidian, or even proper length knives. You could create club like weapons (various maces or bludgeons), spears and arrows. Problem is they would break easily in the contact with enemy armor or his (iron) weapon. In contrast to that, iron is malleable, it could last very long time and you could sharpen it when it gets dull .

The wiki entry on Macuahuitl has an interesting paragraph on their drawbacks, which likely answers your question implicitly:

The macuahuitl had many drawbacks in combat versus European steel swords. Despite being sharper, prismatic obsidian is also considerably more brittle than steel; obsidian blades of the type used on the macuahuitl tended to shatter on impact with other obsidian blades, steel swords or plate armour. Obsidian blades also have difficulty penetrating European mail. The thin, replaceable blades used on the macuahuitl were easily dulled or chipped by repeated impacts on bone or wood, making artful use of the weapon critical. It takes more time to lift and swing a club than it does to thrust with a sword. More space is needed as well, so warriors advanced in loose formations and fought in single combat.

The citation is Townsend, Richard F. (2000). The Aztecs (revised ed.). London: Thames and Hudson. ISBN 978-0500281321. p. 24.

Still, the weapons didn't go away overnight, and were still around a whole generation after Cortez conquered Mexico and introduced steel, if this article on the effectiveness of the macuahuitl (Marco Antonio Cervera Obregón, 2006) is anything to go by:

Its importance as a striking weapon was recorded even in artistic representations in the early time of the viceroy such as the images of the Franciscan temple of Ixmiquilpan in the Stateof Hidalgo, Mexico, which depict a group of Nahua warriors confronting thenorthern Chichimeca armies in the middle of the 16th century (Gruzinski 1994)(figure 12).

This other article on evidence of 17th century maya by Jaime Awe has an interesting footnote (4) that discusses them too, which suggests they were still in use then - if only rarely, one would presume, and possibly chiefly as gifts. The footnote is about the use of the term 'machete' in a contemporary account by López de Cogolludo:

The use of the term machete here is relevant since López de Cogolludo does not use espadas ropeas, the Spanish equivalent of the English rapier. In fact, as far as we can ascertain only two mentions are made to eſpadas ‘swords’ in the whole of the Historia de Yucathan, and surprisingly, both cases refer to Maya weaponry (López de Cogolludo 1688: 2, 6). The first instance is used in reference to Maya warriors, who are said to carry “bows, and arrows, lances, and shields, swords” (p. 2), whereas the second describes the obsidian blade-studded wooden club, known as a maa’k w awitl () in Nawatl (p. 6). As a result, one is left to wonder which term López de Cogolludo would have used to described a rapier, especially considering that machete(s) crop up only six times in the whole Historia, with four occurring precisely in the chapters outlining the Hubelna conflicts (i.e. bk. 11, chaps. 13 & 14, 643, 647, 648, 649).

Similarly, they were apparently still in using atlatls (with stone tipped spears) in the 17th century, at least in Brazil and California.

Aside: irrespective of these drawbacks, Jaguar Warriors were still incredibly fierce if eyewitness accounts of the time are anything to go by:

It is one of the most beautiful sights in the world to see them in their battle array because they keep formation wonderfully and are very handsome. Among them are extraordinary brave men who face death with absolute determination. I saw one of them defend himself courageously against two swift horses, and another against three and four, and when the Spanish horseman could not kill him one of the hors*m*n in desperation hurled his lance, which the Indian caught in the air, and fought with him for more than an hour, until two foot soldiers approached and wounded him with two or three arrows. He turned on one of the soldiers but the other grasped him from behind and stabbed him. During combat they sing and dance and sometimes give the wildest shouts and whistles imaginable, especially when they know they have the advantage. Anyone facing them for the first can be terrified by their screams and their ferocity.

The citation is from Hassig, Ross. Aztec Warfare: Imperial Expansion and Political Control. University of Oklahoma Press (1995). ISBN 978-0-806-12773-6 p. 124, quoting from the Narrative of Some Things of New Spain and of the Great City of Temestitan by an anonymous conquistador.

The Macuahuitl were very sharp, too. Bernal Díaz del Castillo, a conquistador, raised that they could behead a horse in battle (albeit perhaps with some exaggeration):

Pedro de Morón was a very good horseman, and as he charged with three other hors*m*n into the ranks of the enemy the Indians seized hold of his lance and he was not able to drag it away, and others gave him cuts with their broadswords, and wounded him badly, and then they slashed at the mare, and cut her head off at the neck so that it hung by the skin, and she fell dead.

The citation is Díaz del Castillo, Bernal (1956) [ca.1568]. Genaro Garcia (Ed.) (ed.). The Discovery and Conquest of Mexico 1517–1521. A. P. Maudslay (Trans.). New York: Farrar, Straus and Cudahy. ISBN 978-0415344784. p. 126.