I’m aware that the “American” accent dates back to the early 18th century, as noted in another thread about British Colonial America and accents. Did any French speakers in New France have any notably different accents from those of their native France?
I think one part of the best answer to your question might involve interrogating the assumption that, as the accents of the descendants of settler colonists diverge from the accents of their former compatriots, what is happening is necessarily the colonists losing an accent that is maintained in the mother country. It might be more helpful to think of this phenomenon as mutual divergence from a common ancestor. Thus, settler dialects don't meaningfully descend from the modern dialects of their mother countries any more than we descend from the modern apes that we share common ancestors with.
If anything, the opposite is perhaps more often true as colonial and immigrant communities often serve as linguistically conservative reservoirs where historical accents and dialects are preserved. Indeed, as in your example, the English currently spoken in the US is in many respects both closer to the English spoken by Shakespeare and more homogeneous than the English spoken in England today. Similarly, modern Parisian French has diverged from the language spoken in France during the age of French colonialism in North America at least as much as Quebecois and le Français de la Louisiane have.
Additionally, one thing that might be helpful to note is that in the 1666 census of New France, of 3,215 persons listed, only 16 were identified as "gentlemen of means" for their occupation. The French spoken by the settlers even before they left was, for the most part, not at all the kinds of grammar and vocabulary prized by the Académie Française for prescriptive preservation, then or now.
I recommend this article: Michael Friesner, "Une prononciation «tsipéquement» québécoise?: La diffusion de deux aspects stéréotypés du français canadien," The Canadian Journal of Linguistics / La revue canadienne de linguistique, 55(1), March/mars 2010, pp. 27-53, https://doi.org/10.1353/cjl.0.0070 (in French).
It's a historical overview of the phonology of Canadian French and its most distinctive features. The colonists came from several regions, including the West and Île-de-France, some speaking a local dialect/regional language (unhelpfully known as a patois; it's a negative and unclear term that refers both to other languages and to regional varieties of French). In any case, the speakers were able to speak with each other, and in all probability abandoned their local languages and varieties for a more standardized variety that educated speakers used.
The origins of colonists and contacts (and lack thereof) with other groups, that is, with French speakers in Québec and later with France after the end of French rule in North America are two major factors for a distinctive Acadian dialect as well as an eastern dialect of Québécois French that is influenced by the aforementioned Acadian dialect.
However, languages just change over time, and some features, like certain diphthong patterns particularly observable in Canada, seem to have developed independently; Friesner strongly argues that even if we see a feature in a dialect of France that also occurs in Canadian varieties, we can't assume that the two are genetically related.
Friesner cites refers to William Labov's 2007 article "Transmission and diffusion," published in Language (83:344–387) and Gillian Sankoff and Hélène Blondeau's article "Language change across the lifespan : /r/ in Montreal French" (Language 83:560–588) which will give you an idea of how language changes sociologically (particularly bottom-up changes/"changements d'en-dessous," which tend to come from Montréal, the city with the most influence in the region) and over the course of one's lifetime.
So, in sum, it's not really helpful to speak of "losing" accents, because French changed in France and in the colonies, later Canada, though certain features changed at different rates or in different ways. The bibliography of that article will give you some leads on historical linguistics more precisely (but especially Dans Les origines du français québécois, dir. Raymond Mougeon et Édouard Beniak, Sainte-Foy, QC : Les Presses de l’Université Laval), though my suspicion is that a lot of the material is in French, because Canada, as a bilingual country, puts a lot of effort into linguistics research, and publication in French is encouraged.