First, -ment formed nouns in both Old French and Middle English. For instance, consider the etymology for garment (as on Etymonline and crosschecked with the Oxford English Dictionary or OED).
c. 1400, "any article of clothing," reduced form of garnement (early 14c.), from Old French garnement "garment, attire, clothes" (12c.), from garnir "fit out, provide, adorn," from a Germanic source (compare garnish (v.)), from PIE root *wer- (4) "to cover."
Garnement in Old French (a noun) was formed from a verb + -ment. This kind of formation from Old French -ment and Latin -mentum has several examples, including the French-derived words habiliment, abridgement, accomplishment, and commencement, and the Latin-derived words fragment, aliment, and ornament. ("-ment, suffix," OED)
So both French and English allow for the formation of nouns from verbs with -ment. What's remarkable is that, from Old French to now, -ment could also form adverbs from adjectives, and this didn't carry over to English. The Oxford Guide to Etymology (2009) summarizes its origin from the Latin ablative phrase '(adjective) mente,' which in English terms served as an adverbial:
Affixes often result from grammaticalization of adverbs or prepositions. For instance, the adverb-forming suffix -ment in French (and likewise Spanish -mente, Italian -mente, etc.) originates in uses of Latin mens, ment- in the ablative case in collocations such as clara mente 'with a clear mind'. Such collocations showed semantic broadening, e.g. 'clear manner of being or behaving' rather than simply 'with a clear mind', and the pattern became extended to adjectival bases which had no connection with mental activity, ultimately giving rise to a very productive adverb-forming suffix.
Why did the adverb-forming -ment not carry over to English while the noun-forming -ment did? There's no sure answer, but my training as a medievalist makes a hypothesis tempting. There are a few possible suffixes for forming nouns from verbs in each language because the semantic possibilities are broad, so -ion (procession), -tion (solution), -er (bidder), and others all form nouns from verbs with different senses and limitations; -ment fills a place in the Middle English system for aristocratic items as well as functions of government. (See government.) That incorporation fits the history of England in that period, which had French-derived legal institutions and a French-speaking aristocracy. Shortly after that point, the suffix was naturalized into English and widely applied to words irrespective of origin. (An early example is onement, meaning roughly concord or agreement.)
However, one suffix tends to dominate adverb formation in each language: English has the Old English-derived -ly and French has the Old French-derived -ment. The suffixes are so widespread and actively used that there is little reason for the suffix to cross over.