Why is "one" pronounced as "wan", not "oh-ne"?

Etymonline has this:

Originally pronounced as it still is in only, and in dial. good 'un, young 'un, etc.; the now-standard pronunciation "wun" began c.14c. in southwest and west England (Tyndale, a Gloucester man, spells it won in his Bible translation), and it began to be general 18c.

Wiktionary adds:

one and once are pronounced differently from the related words alone, only and atone. Stressed vowels often become diphthongs over time (Latin bona → Italian buona and Spanish buena), and this happened in the late Middle Ages to the words one and once, first recorded ca 1400: the vowel underwent some changes, from ōn → ōōōn → wōn → wōōn → wŏŏn → wŭn.

It is worth noting that one comes from the same source as the indefinite articles an and a (of which an is actually the older form). It is cognate with the Latin unus — whence the French un you mention (and the Spanish un, and the Portuguese um...) — as well as with the German ein, the Russian один, etc.

The word one would be expected to be pronounced /oʊn/ in modern English based on either its spelling or its etymology. The pronunciations starting with /w/ are exceptional. As far as I know, the use of /w/ in this word is not related to French in any way. As RegDwigнt's answer says, French un is similar in form to English one because the two words are cognates.

A 2013 blog post by Piotr Gąsiorowski, "The Secret Ways of Weak Forms: Here Comes a New ’Un", indicates that /w/ may have been inserted at the start of other words beginning with rounded vowels in older stages of English, but that one (and once) is the only surviving example of this phenomenon.

There is spelling evidence to show that some varieties of Middle English tended to insert a glide before any word-initial vowel, as if trying to force every word to begin with a consonant. The glide tended to match the features of the vowel, so we usually find the rounded back glide /w/ before rounded back vowels, including ME /ɔː/. Middle English scribes had no common standard to conform to, so they often used “ear-spellings”, revealing their pronunciation habits. We thus find occasional w-initial variants of words like ǭte ‘oat’, ǭth ‘oath’, ǭn ‘one’, ǭnes ‘once’, ǭnliche or ǭnlie ‘only’, ǭld ‘old’, ǭk ‘oak’, etc. The idealised “dictionary” spellings cited in the preceding sentence represent a Late Middle English “virtual norm” that didn’t really exist. The actual spelling was highly variable, and the documented variants include wote, woth, won, wonys, wonlyche, wolde, wooke, etc.