β is found as early as 2nd century BC per Grundy
Grundy1, who I've just started reading, says that the use of β to transcribe Latin V is attested in inscriptions as early as the 2nd century BC. Furthermore, Grundy says β is found 14% of the time as the transcription of Latin V in manuscripts of Polybius, who wrote in the second century BC.
Grundy gives the following percentages for β as a transcription of V in Greek inscriptions in first two centuries BC and second two centuries AD:
- 2nd Century BC Inscriptions: 27% (only a small sample: 4 with β, 11 with ου)
- 1st Century BC Inscriptions: 34%
- 1st Century AD Inscriptions: 29%
- 2nd Century AD Inscriptions: 33%
He also gives percentages for its occurrence in manuscripts of specific authors, such as Josephus (31%) Plutarch (49%), Dionysus of Halicarnassus (30%) and Diodorus (36%).
There is more data in the paper, which you can read at the link.
I found the data that Grundy reports surprising and I'm not sure why their implications for the timeline of the [w] > [v] change are not discussed more in more recent literature.
I did find the following article that, citing Grundy among others, argues that the change of Latin [w] to [β] had begun by the year 200BC: "The Greek Transliteration and Pronunciation of the Latin Consonant U", by Bradley Buszard, Glotta, Bd. 94 (2018), pp. 109-126.
Kantor (2017 dissertation) shows different frequencies looking at a different corpus
"The Second Column (Secunda) of Origen’s Hexapa in Light of Greek Pronunciation", a dissertation by Benjamin Paul Kantor (2017), contains a section on Latin transcribed into Greek papyri, most of which are "bilingual Greco-Latin
glossaries" (page 134).
The corpora Kantor considers show a much higher percentage of transcription as ου in the first and second centuries AD; in fact, β for V is only found before the 4th century AD in one text (containing three examples) (SB III.I. 6304, page 140).
Kantor says that the low frequency of β is probably related to the genre of the texts in the corpus:
Even though Latin v was pronounced as [β] already from the first or second century CE in Egypt, glossarial and grammatical texts used for language instruction maintained the old pronunciation [w] up until the fourth century CE. During this period, language-learning texts represented Latin v with ου, reflecting an archaic or standard pronunciation, and non-didactic texts represented Latin v with β (e.g., SB III.I. 6304), reflecting everyday colloquial pronunciation. After the fourth century CE, it seems that the pronunciation of Latin v as [β] had become so universal that even in grammatical texts Latin v was represented with β.
I find this explanation quite plausible, since it's not too hard to come up with examples of similar archaisms in transcription or pronunciation of modern languages (such as the fact that "sch" (or French "chtch") is still commonly seen as a transcription of the Russian letter Щ, which was previously pronounced [ɕtɕ] but is now pronounced [ɕː]).
Similarly, Grundy argues that the transcription ου was favored in the spelling of official names due to conservatism well after the appearance of the first transcriptions in β (pages 15-17).
- "The transliteration and pronunciation of the Latin letter V", by G.B.
Grundy, 1907, mentioned in a comment by TKR