As fdb mentioned in a comment on another question, Tibullus references the planetary days in the first century BCE:
Aut ego sum causatus aves aut omina dira,
Saturni sacram me tenuisse diem.
I made excuses [not to leave]: birds, or dire omens, or that I held Saturday sacred.
(I.3.15, translation mine)
So even in late Republican times, it seems the seven-day week had caught on. Early mentions tend to associate it with Judaism (such as Tibullus refusing to travel on shabbath), but the Jewish system numbered the days, rather than naming them.
Then where did the names come from? Cassius Dio associates them with astrology rather than the Jewish shabbath; he traces it back to Egypt, but more likely this came from the Babylonian four-cycles-of-seven-days within each lunar month (since the Babylonians originated the division of twenty-four hours).
τὰς ὥρας τῆς ἡμέρας καὶ τῆς νυκτὸς ἀπὸ τῆς πρώτης ἀρξάμενος ἀριθμεῖν, καὶ ἐκείνην μὲν τῷ Κρόνῳ διδούς, τὴν δὲ ἔπειτα τῷ Διί, καὶ τρίτην Ἄρει, τετάρτην ἡλίῳ, πέμπτην Ἀφροδίτῃ, ἕκτην Ἑρμῇ, καὶ ἑβδόμην σελήνῃ,
If you start enumerating the hours of the day and night, starting from the first one, and assign the first one to Cronus, the second to Zeus, the third to Ares, the fourth to Helios, the fifth to Aphrodite, the sixth to Hermes, and the seventh to Selene,
κατὰ τὴν τάξιν τῶν κύκλων καθ᾿ ἣν οἱ Αἰγύπτιοι αὐτὴν νομίζουσι, καὶ τοῦτο καὶ αὖθις ποιήσας, πάσας τε οὕτω τὰς τέσσαρας καὶ εἴκοσιν ὥρας περιελθών, εὑρήσεις τὴν πρώτην τῆς ἐπιούσης ἡμέρας ὥραν ἐς τὸν ἥλιον ἀφικνουμένην.
according to the order of the cycles that were developed by the Egyptians, and if you do all this again, going through the full twenty-four hours, you'll discover that the first hour of the second day is assigned to Helios.
καὶ τοῦτο καὶ ἐπ᾿ ἐκείνων τῶν τεσσάρων καὶ εἴκοσιν ὡρῶν κατὰ τὸν αὐτὸν τοῖς πρόσθε λόγον πράξας, τῇ σελήνῃ τὴν πρώτην τῆς τρίτης ἡμέρας ὥραν ἀναθήσεις, κἂν οὕτω καὶ διὰ τῶν λοιπῶν πορεύῃ, τὸν προσήκοντα ἑαυτῇ θεὸν ἑκάστη ἡμέρα λήψεται.
And if you do all that again in just the same way for the next twenty-four hours, then the first hour of the third day will be given to Selene, and if you keep going through the remaining days, each and every day will be assigned its respective god.
ταῦτα μὲν οὕτω παραδέδοται.
So, that is the traditional way of it.
(XXXVII.19, translation mine)
In other words, the astrological order of the planets was Saturn, Jupiter, Mars, Sol, Venus, Mercury, Luna—in descending order of distance from earth (in their model). Each hour is assigned to one of the planets, following this pattern; since 24 ≅ 3 mod 7, each day starts three places further around the cycle.
Thus, if you assign each day to a planet based on its first hour, you get the traditional order we all know today: Saturn, Sol, Luna, Mars, Mercury, Jupiter, Venus. The day names were just diēs plus the planet's name in the genitive: diēs Saturnī, diēs Sōlis, diēs Lūnae, diēs Martis, diēs Mercuriī, diēs Jovis, diēs Veneris. (In English: Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday.)
Dio also provides evidence that the system was really a repeating pattern of seven days, not strictly tied to the lunar month as the Babylonian system had been.
οὕτω μὲν τὰ Ἱεροσόλυμα ἐν αὐτῇ τῇ τοῦ Κρόνου ἡμέρᾳ, ἣν μάλιστα ἔτι καὶ νῦν Ἰουδαῖοι σέβουσιν, ἐξώλετο.
And so Jerusalem was destroyed right on the day of Cronus, which even now the Jews mark as holy above all other days.
(Epitome of LXV, 7.2, translation mine)
The Jewish shabbath still lines up with our Saturday, which implies there haven't been any extra days added to follow the lunar month (since it's known that the Jewish week has always been seven days without adjustments).
Were these names "official"? Probably not—as in, the government wasn't regulating them. Rather, they were de facto names invented by the astrologers, and some celestial pattern or other kept them in sync across the world. For comparison, people nowadays generally know what zodiac sign they were born under, even though no government or standards organization legislates that: they're just determined by the positions of the stars.
The official system, then, remained the Nundinal cycle up until the time of Constantine (who made the seven-day cycle official, renamed diēs Sōlis to diēs Dominī, and put it at the beginning, with the Jewish shabbath at the end).
But the names in this system were far less exciting. The eighth day of every cycle was the Nundinae, always plural, and that was the only day with a specific name; other days were marked relative to them, like with the Nones, Ides, and Kalendae. (That is, the day after the Nundinae had no special name apart from postridiē Nundinās.)
While the Nundinal cycle was official up until Constantine's edict, it doesn't seem to come up much in post-Republican literature. Cicero complains about people not respecting the Nundinae like they're supposed to, but after the Julio-Claudian period, I've only found passing references in histories. For all practical intents and purposes, the seven-day planetary cycle had caught on, and was the de facto standard for anything that required weekdays—the Nundinal cycle, meanwhile, died a quiet and unremarkable death.
