The answer is a lot more recently than you'd think and also, it depends. You've actually asked two questions here, 1. when did the Romance languages stop calling themselves Latin? and 2. At what point did Spanish, French, and Italian start calling themselves español, français, and italiano? Because these are separate developments. Let's tackle the second question first.

Part I: National Languages

You may notice the names español, français, and italiano line up very nicely with the names of the countries España, France, and Italia. This is not a coincendence. The use of these names was closely tied to the development of the modern European nation-state and the eradication of other regional languages. Even today, if you go to Spanish, Italy, or, to a much lesser degree, France, you will likely run into what locals often call dialects, but are in reality distinct Romance language varieties; the European context, is, after all, the birthplace of the phrase "a language is a dialect with an army and a navy".

In the Spanish and Italian context, these names were applied to regional varieties, Castellano and Toscano/Florentino as these regions began to establish a larger territory, politically and socially dominated from the center. In Spanish, in fact, the name español is so new that it is actually a loan, likely from Catalan or a similar language; if it were a native term, we'd expect españuelo as a natural outcome.

In both places, this was a slow process that sped up at certain moments, in Spain, the marriage of the Catholic monarchs in the late 15th century was the turning point, but it wasn't until 1923 that the Real Academia Española began to refer to the language consistently as español instead of castellano. And throughout the Spanish-speaking world, castellano is still preferred in many places. Surprisingly enough, if not for the success of the Spanish Academy in unifying the various national academies of Spanish-speaking America, we might think of Spanish, Chilean, Argentine, Mexican, and Colombian as closely related but distinct languages.

In Italy, codification of a Tuscan-based dialect began during the rinascimento, the notable figure here was Dante Alighieri whose writings became the cornerstone for the court-language throughout the Italian peninsula during the early modern era. When Italy was unified in the 19th century, this language became tied to a broader Italian identity and when unification was complete, it became the official language of the new country.

France is a bit more complicated as unlike Spain and Italy, which were geographic labels that were used to assert a national unity over that territory, France was tied to the ethnic identity of the Franks, a Germanic people who gained control over northern Gaul and the Low countries after the fall of the Western Roman Empire. This label went through many many many permutations, ranging from applying to only these Germanic conquerors to their empire, which came to dominate nearly all of Western Europe and then to a monarchy based in Paris. The language variety that came out of Paris thus became the dominant language as that monarchy established dominion over the territory we think of as France now.

In all these cases, key steps were the development of language academies, the Accademia della Crusca, founded in 1583 in Florence, was the first language academy. It was followed by the Académie Française, founded in 1635, and the Real Academia Española, founded in 1713. In all cases, these academies were backed up by rulers; della Crusca had Cosimo I di Medici and l'Académie Française had Cardinel Richelieu and King Louis XVIII. Whereas, in the Spanish case, it's not called the ROYAL Spanish academy for nothing, it was one of King Felipe I's pet projects. Felipe was the first Bourbon king and famously centralized power in the Spanish empire. Funnily enough, he was a native French speaker.