I can only comment for French, but it must have been before the Council of Tours of 813, where it was decided that churches would stop giving homilies in lingua latina and start giving them in linguam Romanam rusticam 'the rustic Roman language* (or in tudesque the Germanic dialects of Charlemagne's empire, but that doesn't bear on your question).

Source: Jacques Chaurand's Nouvelle histoire de la langue française, pp. 27ff.

From what I recall from my course on the history of French as an institution, one of the big reasons they did this was because people used to go to church, but wouldn't pay attention, because they didn't understand what was being said at all. They would just hang out and chitchat during the mass itself. The decision of the Council of Tours of 813 was thus supposed to reduce the egregious disregard for the service itself.

This is a tricky question because the premise itself is wrong. PLOT TWIST: Latin is still spoken throughout the Romance-speaking countries.

Except it is not the Latin you think of when you think of Latin.

What we usually call Latin is more appropriately called Classical Latin, in order to make a distinction between it and other forms of the language spoken by the Romans, their subjects, and the people that kept living in the territories that were once Roman provinces. Classical Latin is, for all intents and purposes, an artificial language, not exactly a conlang, but a language meant to represent the "purest" form of Latin, ie one that nobody actually spoke as a native language. A language almost confined to Academia.

Vulgar Latin, on the other hand, refers to the language of the common people that inhabited the Republic and afterwards the Empire. Vulgar Latin is a distinct form of Latin of which little written record remains to this day. Vulgar Latin, as opposed to Classical Latin, was the every day language of the people of the empire. As you can imagine for a society of such a stretching across three continents, with no access to mass-media, the Vulgar Latin of Hispania (modern day Spain) would be different from the Vulgar Latin of Gaul (modern day France), both of which in turn would be different from the Vulgar Latin spoken in Dacia (modern day Romania). While the mutual intelligebility of each dialect of Vulgar Latin might have varied across the empire, all of the provinces, except maybe Greece, Judaea, and Egypt, spoke more or less the same brand of Vulgar Latin.

This gradual drifting between the dialects of Vulgar Latin did not begin with the collapse of the Western Roman Empire, but must have begun as soon as the empire grew as big as it would. After the fall of the Empire, though, the influence of Rome ceased to be all-present in Gaul, Hispania, Romania, and Italy, and each of their own dialects of Vulgar Latin survived, more or less isolated, and went through a continuum of changes that resulted in modern day Romance languages, which, while undoubtedly related to Classical Latin, are not exactly descendants of it.

TLDR What you call Latin is actually Classical Latin and was never actually spoken in any of the places where a Romance language occurs.

The question of when the people speaking these Romances stopped referring to their language as Latin (Vulgar or otherwise) is also problematic because you can't really tell when every single one of the speakers just began knowing their language wasn't Latin anymore. This is why the oldest records of a given Romance are important, because they give us an idea of the time at which whatever was written was not reminiscent of Latin anymore.

In my opinion, the most interesting case of this hypothetical consciousness of knowing your language is no longer Latin comes from the earliest example of an Iberian Romance, the Glosas Emlianenses. The Glosas are a cute example of somebody struggling with a language that is not their own: they are margin notes written in a sort of Proto-Spanish by a monk who was trying to translate a Latin text. This significance of these documents is the certainty that Latin was no longer intelligible to this monk. It follows that he was perfectly aware that his language, whatever it was, was no longer Latin.

I wouldn't be surprised if self-awareness of the differences actually predated many of these changes. In fact, one could argue that language groups on the edges of the Roman empire never actually spoke what they perceived as Latin, but did speak a sort of contact language which was heavily latinized, and was to some extent mutually intelligible. Those who had a "proper education" would have been able to write (and to an extent speak) at a more formal register, which would fit the standards of Latin, but the uneducated masses might never attain this proficiency.

This is mostly by analogy of the example of Arabic, which likely provides some elucidation. Across the Arabic-speaking world, people refer to the formal register of the language as "Arabic" and their own language differently. That isn't to say that they would not call their own daily speech Arabic in contrast to French or Chinese, for example, but in their common self-perception, it is the "other" more formal register that is Arabic, whereas their own language is slang (often using a different verb rather than speak even). Some Arabic-speaking countries wouldn't make this distinction, but many would. There is even a joke in the Levant, that someone who speaks formal arabic is "speaking Mexican" because of the dubbed-over Mexican telenovelas that have been popular, and used a very formal register of speech.

TL;DR it all depends on context, but in some contexts, it is possible that the majority of people in Gaul, Iberia, etc never perceived themselves to "speak Latin" despite being able to communicate with others in a mutually intelligible variant of Latin.

PS Further proof would be in the continuity of Italian dialects. Calabrian, for example has continued to be distinct from standard Italian since pre-Roman times

Nicholas Oster's Ad Infinitum: A Biography of Latin is a worthwhile read if you're interested in this. He points out two big moments in the 8th/9th centuries. One was Alciun's reforms of Latin to attempt to make the spelling and pronunciation consistent both throughout the Carolingian empire and with what was known of Roman Latin. As Ostler puts it:

One effect of Alcuin's reforms must ultimately have been to impress on everyone that Latin, as written and spoken, was actually now a foreign language, not just the written, quasi-eternal form of Romance speech.

Another effect was reformation of the language used in church services, culminating in translation into the vernacular.

With each priest following his home pronunciation, it was possible - at least in Romance-speaking countries - for the Latin text to have been read pretty much in line with the local language, hence understood by those who could not read. The newly antiquated, universal Latin, by contrast, was a foreign language everywhere, accessible only to those who had studied it. So quite soon after Alcuin's reforms, rulings were needed to guarantee that Church services would still make sense. At the Council of Tours in central France in 813, as at the Council of Mainz in Germany in 847, an explicit exception was made, to guarantee the continued understanding of the countryfold: "...and that each should work to transfer the same homilies into rustic Romance or German language [rusticam romanam linguam aut thiotiscam], so that all can more easily understand what is said."

That said, this is a generalization of all of Roman Europe. Certain dialects, e.g. Provencal, developed separate literature and identities sooner than others. One of the reasons we talk about Dante and the birth of Italian lit in the same breath is because it was Dante's explicit intention to develop a literary version of Italian Romance. In the early 1300s he wrote about the relationship between Romance and Latin and the need for a prestige variety of Romance in De Vulgari Eloquentia - though, of course, he wrote about it in Latin.

Regarding your last question, another concern is that language names and identities are shaped by political circumstances. "I speak French" really means "I speak the same dialect as the King of West Fancia". "Spanish" is the language of the Kingdom of Spain, and was "Castillian" back when it was the Kingdom of Castille. There are plenty of other languages in Iberia, e.g. Galician, Asturian, Aragonese, but they've been slowly losing ground to Portuguese and Spanish, which have nation-states to back them up. Both France and Spain have, in the past, tried to stomp out regional languages, with varying levels of success. In Italy, as in Germany, regional languages thrived through the 19th century, but have been strong deprecated in favor of the national language since unification. Many of them (e.g. Corsican, Piedmontese) are as good as dead, while Sicilian, Neapolitan, and Venetian still boast several million native speakers each.

Another good source is Roger Wright's Late Latin and Early Romance in Spain and Carolingian France, though it's obviously restricted to those two countries.