Is it taken for granted now that the Latin v was pronounced like the modern English w?

That PerClupi's text transcribes the Latin uideo as opposed to the more often seen video suggests that it is.

Intriguingly, the English word for fermented grape juice is (I think) the only modern cognate that retains the w sound from the original Proto-Indo European *win-o-.

Gothic has often been quoted to establish that c was a surd [i.e. voiceless] guttural; for example: Latin, lucerna -- Gothic, lukern; Latin, acetum -- Gothic, akeit; Latin, carcer -- Gothic, karkara.  Its testimony might with equal propriety be taken in regard to Latin consontal v.  In borrowed words this letter is always transliterated by the Gothic w, which had the same sound as English w (Browne's Goth. Gram. p. 19).  The following are examples:--

	Latin,	vinum.		Gothic,	wein.
	"	evangelium.		"	aiwaggéli. [ * ]
	"	oleum.		"	olêw.
	"	cautio.		"	kawtsjo.

The evidence which may be drawn from Anglo-Saxon in regard to the sound of consonontal v in Latin impresses me as strongly as that from Gothic, and yet it has never been quoted so far as I know.  There are three words in Anglo-Saxon which were borrowed from the Romans before 500 A.D.  These are win, wic, and weall, from Latin vinum, vicus, and vallum respectively.  The form of these words shows that the Latin v was w when this borrowing took place.

There can be no reasonable doubt that these are borrowed words (Curtius, I. 487; Skeat's Principles of English Etymology, I. 398).  Nor can it be urged, as might be done in the case of c, that the Goths and Saxons had no other letter to represent the Roman v.  The had the surd spirant f, which certainly might have been used, and probably would have been in some cases at least if the Latin v had not been distinctly like w.

JSTOR: The Classical Review, Vol. 7, No. 1/2 (Feb., 1893 ), pp. 5-7
On Some Neglected Evidences of the Sound of C, V, and S in Latin
J. C. Jones The Classical Review, Vol. 7, No. 1/2 (Feb., 1893), pp. 5-7
Published by: Cambridge University Press on behalf of The Classical Association

Bold and bracketed text mine.  In particular, I asterisked aiwaggéli because in the original text, it is spelled aivaggéli.  However, guessing that this was a typo, I checked and confirmed that aiwaggéli is indeed an alternate and perhaps antecedent spelling for aivaggéli, and most likely the one that Professor Jones intended to use to make his point.

Also, curious about the letter VV, surely enough, Wikipedia had something interesting to say about it:

History

The earliest form of the letter W was a doubled V used in the 7th century by the earliest writers of Old English; it is from this <uu> digraph that the modern name "double U" comes. This digraph was not extensively used, as its sound was usually represented instead by the runic wynn (Ƿ), but W gained popularity after the Norman Conquest, and by 1300 it had taken wynn's place in common use. Other forms of the letter were a pair of Vs whose branches cross in the middle. An obsolete, cursive form found in the nineteenth century in both English and German was in the form of an "n" whose rightmost branch curved around as in a cursive "v" (compare the shape of ƕ).

The sounds w (spelled with U/V) and b of Classical Latin developed into a bilabial fricative β between vowels, in Early Medieval Latin. Therefore, V no longer represented adequately the labial-velar approximant sound w of Old High German. In German, this phoneme w later became v; this is why German W represents that sound. In Dutch, it became a labiodental approximant ʋ (with the exception of words with EEUW, which have eːw), or other diphthongs containing -uw.

Usage

There are only five major European languages that use W in native words: English, German, Polish, Dutch, and Welsh. English uses it to represent w, German and Polish use it for the voiced labiodental fricative v (with Polish using Ł for w), and Dutch uses it for ʋ. Unlike its use in other languages, the letter is used in Welsh to represent vowels as well as consonants. English also contains a number of words beginning with a w that is silent in most dialects before a (pronounced) r, remaining from usage in Anglo-Saxon in which the w was pronounced: wreak, wrap, wreck, wrench, wroth, wrinkle, etc. (Certain dialects of Scottish English still distinguish this digraph.)