With the help of Wiktionary, we know two useful Midlle English etymologies of the word "bow".
From Old English boga, from Proto-Germanic *bugô. Cognate with Dutch boog, German Bogen, Swedish båge, Danish bue.
From Old English būgan, from Proto-Germanic *beuganan. Cognate with Dutch buigen, German biegen.
And the change from "g" to "w" here is very interesting to me, but I am unfamiliar with the process, even the name of the "sound law".
There is probably some phonological rule at work where g at the end of a root disappeared in English. The w in bow is probably best seen as a compensatory change in vowel sound. The phenomenon can be observed in English/Dutch low/laag, lay/leg-, lie/lig- (lie on the ground), lie/lieg- (tell a lie), weigh/weeg-, way/weg, saw/zaag, high/hoog, neigh/neig, bow/boog (and arrow), bow/boeg (ship) etc.: you can see how the g disappears in English in many words.
The Old English g in these words evolved into a labialized velar glide [w] before being finally lost in the modern English forms. Here is a quotation from Donka Minkova's A Historical Phonology of English (2014), §7.4 "The diphthongal system of ME":
The voiced velar fricative [ɣ] in OE was always preceded by a back vowel. The further lenition of [ɣ] to [w] and subsequent merger under [w] was under way in late OE/early ME, evidenced by variant spellings such as gnagan ~ gnawen ‘gnaw’, lage ~ lawe ‘law’, boga ~ bowe ‘bow’. When the approximant [-w] appeared in the coda, increasingly likely at a time of general weakening of the final unstressed vowels, it was vocalised and the glide became the second element of a diphthong: OE bo.ga ~ bo.we (with possibly ambisyllabic intervocalic consonant) > ME bow(e) > bow [bow] ‘bow’, OE būgan, ME bouen ~ buwe(n) [buwən] ‘bow down’.
Old English g between vowels is reconstructed as being pronounced as [ɣ] (a voiced velar fricative). This is very similar to (or even practically indistinguishable from) [ɰ], a voiced velar approximant, and if you add rounding to [ɰ], you get [w]. So even though Old English g contrasted with /w/, the contrast depended mainly on the single feature of rounding. By middle English, [ɣ] after a back vowel had apparently rounded, thus merging with /w/.
Whenever w came to be at the end of a syllable it was not preserved as a consonant phoneme in present-day English, but we can tell that [w] was formerly present in such words by the effect that it had on the quality of the preceding vowel or diphthong that it merged with, as shown by gnaw /nɔː/, from Middle English gnawen [ˈgnawən] (also gnaȝen), from Old English gnagan [ˈɡnɑɣɑn].
In the case of words with Old English /og/ such as boga, we know that the /g/ was not just lost with compensatory lengthening of the preceding vowel because there are obscure accents of modern English that pronounce (or pronounced) words like toe and tow differently. See the section "toe–tow merger" in the Wikipedia article "Phonological history of English diphthongs":
The toe–tow merger is a merger of the Early Modern English vowels /oː/ (as in toe) and /ou/ (as in tow) that occurs in most dialects of English. (The vowels in Middle English and at the beginning of the Early Modern English period were /ɔː/ and /ɔu/ respectively, and they shifted in the second phase of the Great Vowel Shift.)
The merger occurs in the vast majority of Modern English accents; whether the outcome is monophthongal or diphthongal depends on the accent. The traditional phonetic transcription for General American and earlier Received Pronunciation in the 20th century is /oʊ/, a diphthong. But in a few regional accents, including some in Northern England, East Anglia and South Wales, the merger has not gone through (at least not completely), so that pairs like toe and tow, moan and mown, groan and grown, sole and soul, throne and thrown are distinct.
In 19th century England, the distinction was still very widespread; the main areas with the merger were in the northern Home Counties and parts of the Midlands.
The distinction is most often preserved in East Anglian accents, especially in Norfolk. Peter Trudgill[9] discusses this distinction, and states that "...until very recently, all Norfolk English speakers consistently and automatically maintained the nose-knows distinction... In the 1940s and 1950s, it was therefore a totally unremarkable feature of Norfolk English shared by all speakers, and therefore of no salience whatsoever."
In a recent investigation into the English of the Fens,[12] young people in west Norfolk were found to be maintaining the distinction, with back [ʊu] or [ɤʊ] in the toe set and central [ɐʉ] in the tow set, with the latter but not the former showing the influence of Estuary English.
Walters reports the survival of the distinction in the Welsh English spoken in the Rhondda Valley, with [oː] in the toe words and [ou] in the tow words.
Furthermore, consider Middle English spelling. The OED records the following spellings in Middle English of the descendant of boga: boȝe, bou, bowe, bouwe, boghe. There is alteration from a spelling with g or ȝ to a spelling with w, which fits well with the hypothesis of a change of the consonant [ɣ] to /w/ (or a non-syllabic vowel or offglide [u̯]).
In contrast, the OED records no Middle English spellings with ȝ or w for the descendants of Old English tā (toe).
In the case of būgan, after the original [ɣ] became rounded the resulting /w/ probably merged sooner with the preceding vowel because [w] is basically just the non-syllabic version of [uː]. As Alex B. wrote, the ow in bow from būgan is a digraph representing the outcome of the Old English long vowel ū: it can be found in words that never had w or g such as now from Old English nū.
Your question is based on a wrong assumption - the letter w in bow2 is due to a spelling convention imposed by French scribes. At first, long u was respelt as ou (like in house), and then all word-final ou's were respelt as ow - I'm sure you've already noticed how rare or non-existent English words are that end in ou. Then long u was diphthongised - the Great Vowel Shift. This stuff is pretty basic, and you can read about it in any intro textbook on the history of English.
/aʊ/and/oʊ/). It's a bit like asking if "h" is pronounced in "the". Now a good question is how this spelling came about and whether it is due to once pronouncing the "o" and "w" literally and separately.