Interesting question. As far as I can tell, at least parts of the treasure ended up with Jean (Jehan) Ango of Dieppe, Jean Fleury's patron and other parts with Francis I. Items from the Aztecs were on display in Jean Ango's mansion for several years. Whether they made up the Aztec's greatest treasure is another question - I'll first look at accounts of the treasure and then turn to its French fate.

I.

The first of two accounts I will mention comes from Hernan Cortés. He mentions this treasure being taken after the fall of Tenochtitlan, because he had lost earlier treasures after the Spaniards had left Tenochtitlan during the Noche Triste.

after the Royal fifth had been taken from the gold that was melted down, we agreed that all the jewely pertaining to my companions and myself should be sent to your Highness, and their worth was no less than that of those which we captured the first time [before the Noche Triste]. With all hast and every care I dispatched them, together with three thousand pesos in gold bars in the charge of Julián Alderete, who at that time was Your Majesty's treasurer; but they were seized by the French. [Cortés, 440]

Cortés goes on to blame not himself but the those who "failed to provide an escort from the Azores". He also mentions sending another shipment with a silver fieldpiece with Diego de Soto, leaving Cortés without "a single peso de oro". When talking about the treasure's loss Cortés remarks that

I was much grieved at their loss, for they were all so rich and strange that I greatly desired Your Majesty to see them, for, besides the benefit which Your Majesty would have gained from them, my services would be more manifest. I am not altogether sorry, however, that they were lost, for Your Majesty would have little need of them, and I shall endeavor to send others, richer and more wonderful, according to the news that I have of some provinces which I have lately sent to conquer, and others to which I will send as soon as I have the men. [Cortés, 330]

We should note that Cortés mentions these riches in order that ask Charles V. reimburse him for all his expanses spent during his campaigns. So he had an interest to both show how he gave the maximum of riches to the Spanish Crown; and that the shipment's loss was not a problem because he could find greater riches still. Actually, after the fall of Tenochtitlan Cortes' men had expected more gold as reward for their services. This in tandem with the payments to the Crown in part explains Cortés further campagins to Southern Mexico shortly afterwards.

Here's the second account by the conquistador Díaz de Castillo, writing decades after the conquest period - in contrast to Cortés writing more contemporaneausly:

[Cortés] gave them [Alonso Davila and Antonio de Quinones] two ships with 28.000 castellanos in gold bars and with the clothes of Moctezuma for the emperor [Charles V.]. These clothes were an especially precious gift worthy of our emperor. It contained many jewels and gold workings, pearls as big as hazel nuts, and other riches. He added to them some of the giants' bones we had found in the temple of Coyohuacan. Furthermore three jaguars and other rarities came on board, which I cannot list anymore today. [Díaz, 447-48, my transl.]

In more detail he also mentions the accounts of the conquest campaigns and of Mexico given by the conquistadors, which were important additions to the items. Díaz de Castillo goes on to recount the treasure's capture by Fleury. According to him, prior to the capture the jaguars had escaped and wreaked havoc on board; and the second captain Antonio de Quinones got in a fight over a girl and was killed, leaving Alonso Davila as the sole commander. If this is only an example of Díaz' penchant for telling a good story is hard to say.

Díaz was more removed in time from the events thant Cortés, and his main interest was not to receive further reimbursements but rather to show the important role other conquistadors had played in the campaigns which he felt had been belittles by Cortés. So Díaz admitted more freely that the shipment's capture had meant a great loss of, we can infer, a great treasure.

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II.

Jean Fleury had been employed by Jean Ango, a Norman ship-owner who would finance exploration missions. In exchange for the use of his large fleet of ca. 70 ship, Ango would be awarded noble titles including Viscount and later captain of Dieppe by the French king Francis I. Dieppe was a port city of major importance to France at the time. In effect, the king gave legal permission to Ango to use his ships for privateering as was common at the time. He also maintained that seizing these Spanish ships was a legitimate act of war, since France and Spain had been at war on and off since 1494; and since France had been left out of the Treaty of Tordesillas (which had divided the world into Spanish and Portuguese spheres of influence with the pope's blessing). The first of these piracy actions was Jean Fleury's capture of the two Spanish ships, forming the basis for Ango's riches and high position. It also signaled the beginning of a long tradition of French, English and Dutch pirates capturing Spanish cargo returning from the Americas; and led to the Spanish Crown's large-scale reinforcement of many fortifications, esp. in the Carribbean.

Díaz de Castillo has once more a more detailed account of the capture than Cortés. He notes how the pirate took over both ship near the Island Tercea beyond the Acores; and shortly afterwards captured a third Spanish ship coming from San Domingo with further riches.

With these precious prices the pirate returned to France a rich man and offered rich presents to the French king and the Admiral of France. All of France marvelled at the treasures, which we had sent our emperor. At that time the king of France let it be known to the emperor [Charles V] that he could finance the war with him only with the riches he received from the West Indies. [Díaz, 450]

Díaz also notes how Jean Fleury would capture some more Spanish ships only to be hanged on the emperor's orders. The "All of France marvelled at the treasures" implies already that the treasures were on display somewhere. Turning now to secondary literature, according to Nigel Cawthorne

the first wealth of the New World went, not to the court in Madrid, but Fleury's patron Jean Ango of Dieppe. Five years later people came from as far away as Paris to witness a masque Ango held at his new manor in Vangeville outside Dieppe where the players paraded the finery of the now defunct Aztec empire.

Ango was also known as the "Medici normand" in connection with his large Renaissance villa built from 1533-1540 at Vangeville. He was visited there by Francis I., and stored many of his trasures and of the knowledge he attained from later explorations and privateering actions there. Michael Wintroub notes that the information stored on board was probably more useful to the French than the material gains:

Indeed, gold, jewels, and curiosities were not the only items of value to be found on Cortés' ships -- they also contained detailed reports of his exploits and information about the riches to be found in the New Worlds he had conquered. This information was found not only on paper, but also in the experience of the ship's captured crews, as for example, Alonso de Ávila, Cortés bookkeeper, who was to be held prisoner in France for the next three years.

These "detailed reports" bring to mind the accounts given by the conquistadors, as mentioned by Díaz de Castillo . In addition to the conquest campaigns Díaz also mentions administrative changes following the conquest; later military campaigns; and that the conquistadors asked Charles V for high positions for Cortés and themselves. Overall there would have been quite some relevant information for both the French king in planning French attacks on Spanish ships; and for Jean Ango and others in planning later exploration missions to the Antilles. Jean Argo would soon after launch expiditions: among them those of Giovanni da Verrazzano in 1524 of the zone between Newfoundland and Florida; and of Aubert to Newfoundland.

Wintroub gives a fascinating description of Argo's mansion as a hub of intellectual and scientific activity at the time (which goes beyond this question's scope). However, in later years Argo's ventures went out of luck and he would fall out of favor with the later king King Henry II. His forced participation in a royal armament project led to his near bankruptcy and then imprisonment after 1549. I have not found anything else on where the Aztec treasures went from Argo's possession, and about those held presumably by the French king - hopefully others more familiar with French history can add to it. However it seems certain that at least parts of the treasure were put on display for a number of years, with Díaz who wrote his Historia verdadera in the mid-16th century noting a bit hyperbolically how all of France had witnessed them.

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Note: Most books would mention Fleury's capture of the ships without talking more about the treasure's later destination. Then again, even for those items (and native people) brought "officially" by the Spanish to Europe in the early 16th century, it can be difficult to trace them over a longer time-frame -- unless they ended up in a well-known institution or museum at some point.

• Hernán Cortés: Letters from Mexico, ed. Anthony Pagden, Yale UP 2001.

• Bernal Díaz del Castillo: Die Eroberung von Mexiko, Insel Verlag 1988 (only had the German version at hand)

• Michael Wintroub: The Voyage of Thought: Navigating Knowledge across the Sixteenth-Century World, Cambridge UP 2017

• Nigel Cawthorne: A History of Pirates: Blood and Thunder on the High Seas, Arcturus Pub. 2011.

Edit: Added a 2nd part