Disclaimer: As has been repeatedly pointed out, this is a gradual shift that cannot really be pinpointed. Moreover, in my opinion, it hugely depends on how one interpret any of the several parts in this question.
Duringthe Principate period (27 B.C. – A.D. 284), emperors carefully maintained the façades of Republican government.
The senate continued its legislative, religious, diplomatic and judicial functions in the Principate ... The senate had a role in the legitimization of the emperor; both Trajan and Hadrian wrote to ask the senate to ratify their position. Galba also refused Imperial titles until conferred by an embassy from the senate. This is the paradox; the princeps held supreme power, yet some still insisted on acting as if the senate was of legal importance.
- Wilkinson, Sam. Republicanism during the early Roman Empire. Bloomsbury Publishing, 2012.
Of course, it is obvious to us now that the republic had fallen and a monarchy had taken its place. Yet we should not project modern perspectives onto the ancient Romans.
To be sure, the State was organized under a principate - no dictatroship or monarchy. Names did not matter much ... The Roman, with his native theory of unrestricted imperium, was familiar with the notion of absolute power. The Principate, though absolute, was not arbitrary. It derived from consent and delegation; it was founded upon the laws. This was something different from the monarchies of the East. The Romans had not sunk as low as that. Complete freedom might be unworkable, but complete enslavement was intolerable.
- Syme, Ronald. The Roman Revolution. Oxford University Press, 1963.
Part of the fiction of republicanism was the theoretical authority of the senate to elect emperors. Again, we recognise this to be rubberstamping a late emperor's choices and/or the army's decision. However, again, appearances were nonetheless kept up.
Originally the Emperor was elected either by the Senate or by the people, i.e. the army. Later this practice was modified, but not discernibly changed. In the early period, in the event of election by the army, the Senate at least recognized the army's choice. Under Gaius the day on which he had been proclaimed Emperor by the Senate was celebrated, and even Vespasian was recognized by the Senate ... by the third century, however, we observe recognition by the Senate being dispensed with.
- Mommsen, Theodor. "A History of Rome under the Emperors, ed. Thomas Wiedemann, trans. Clare Krojzl." (1996).
Although the Empire had superseded the Republic, the state continued to maintain an illusion of constitutional continuity. It maintained republican daily activities and its rulers (for the most part) avoided monarchical pretences. That the republic had became a monarchy did not mean the Romans were obliged to realised what had happened to their ancient freedoms. While we know it had essentially became a monarchy, I would argue that the situation was familiar enough for a contemporary Roman to hold onto the fiction of a republic.
Nonetheless, realistically, public perceptions must have begun waking up to reality. This certainly happened by the time of the transition into Dominate, from the Crisis of the Third Century onwards. Unlike the previous era, Emperors abandoned the pretensions of republicanism. Diocletian's formal adoption of monarchical styles definitely could not have gone unnoticed.
Another point: I would argue there was a certain degree of doublethink going on during the late empire. Even while acknowledging the monarch and empire, Romans continued cling to the fantasy of a republic. When the Western Roman Empire finally fell in 476 A.D., the Senate sent the imperial regalia to Constantinople along with a delegation. The deputies carried with them an "unanimous" message addressed to the Eastern Emperor Zeno, in which the Senate:
[S]olemnly "disclaim the necessity, or even the wish, of continuing any longer the imperial succession in Italy; since, in their opinion, the majesty of a sole monarch is sufficient to pervade and to protect, at the same time, both the East and the West. In their own name, and in the name of the people, they consent that the seat of the universal empire shall be transferred from Rome to Constantinople; and they basely renounce the right of choosing their master, the only vestige that yet remained of the authority which had given laws to the world. The republic (they repeat that name without a blush) might safely confide in the civil and military virtues of Odoacer; and they humbly request, that the emperor would invest him with the title of patrician, and the administration of the diocese of Italy."
- Gibbon, Edward. The history of the decline and fall of the Roman Empire. Vol. 6. J. & J. Harper, 1829.