That is a question we don't have a definite answer to. It is certain that by the end of Augustus' rule Latin was the dominant language of the peninsula, though that still does not mean that it was absolutely everyone's mother tongue.

Other than that there are a wide range of estimates. The traditional view, which I guess can be traced back to Mommsen in this case, is that local languages were eroded throughout the II. century. There were multiple contributing factors in that.

One was that Latin colonies were all over the peninsula by then, and thanks to economic factors many of them attracted immigrants from nearby non-Latin states. We also know that Rome itself attracted many immigrants, though time to time these tended to be expelled. But presumably they could learn Latin while living in the city, maybe their children even grew up as bilinguals.

As Italian allies were compelled to send troops for Rome, and these troops served alongside Roman and Latin units, soldiers could learn Latin during their military service.

These are mostly speculations though. Henrik Mouritsen examines the question in his Italian Unification rather closely, and he asserts that Italia was still a linguistically diverse land throughout the II. century, and local languages were still going strong by the end of the century.

However, the Social War changed that. After the Italian allies lost, many of them were punished. Besides losing a substantial part of their population, lands were also taken from them as punishment, and new colonies were founded on these lands. This remained a standard practice through the subsequent civil wars. The winner side tended to punish the cities that chose to support the other side, and rewarded its soldiers with lands confiscated from the losers.

Also, the Italian allies rapidly lost their independence as the Roman Commonwealth was restructured in the aftermath of the Social War. Their soldiers did not necessarily serve in their own distinct units anymore, but, as they themselves became Roman citizens, they would be drafted into the Roman legions. Latin would be more often used as the language of governance, and also would gain additional economic importance.

So, to reiterate, Mouritsen's opinion is that Oscus, Umbrian, Etruscan and other local languages were mostly replaced by Latin in the I. century BC, thanks to intensive colonisation, political pressure, and partially to genocide. Meanwhile the traditional view was that the shift happened earlier, and by the end of the II. century the Italian allies have essentially seen themselves as Roman enough to demand Roman citizenship.

Also worth mentioning that Syme in his Roman Revolution describes Octavian's party as it was mainly composed from the disgruntled elements that were on the losing sides of previous civil wars. Thus, Octavian's coup d'êtat is essentially seen by Syme as Italian nobility and Roman yeomen taking over the government. As Octavian was a moderately wealthy noble from the countryside, Maecenas belonged to ancient Etruscan nobility, while Salviedenus and Agrippa had even humbler origins then Octavian, this view is certainly attractive to me.

Syme also thinks that the oath of allegiance in 32, and the War of Actium is the point when Italia effectively became a single nation. If we accept this statement, we also might conclude that this new national identity could have helped to spread the Latin language, and would have further eroded the position of other languages.