Some common general references on the History of French language are Mireille Huchon's Histoire de la langue or Le français de la Renaissance. In short, yes, the accent would have been very different. First, the Dauphine would definitely have spoken French (which was not a first language for the vast majority of people in the Kingdom of France), as it was the language spoken by the court, but it wouldn't have sounded anything like contemporary French.

What you call the "Old English accent" is most commonly Early Modern English, which is still understood today because of two huge cultural references of the XVIth century, namely the King James translation of the Bible and Shakespeare's work, which are still popular today. The language spoken in the XVth century in France is what we usually call Middle French, and is closer to modern versions of French than Ancient French, but still not very well codified in the time period (remember printing wasn't a thing yet, and manuscripts costed a fortune).

The main differences would have been lexical, but let's skip these and think more about how it would have sounded. First, the r today are pronounced [ʁ], what linguists call a voiced uvular fricative, while the Middle French version would have been a [r] like in Spanish, what linguists call a voiced alveolar trill. The vocalic system would have been more differentiated than contemporary French (more like in Canadian French), as the nasals (an /ɑ̃/, en /ɑ̃/, in /ɛ̃/, on /ɔ̃/, un /œ̃/) would have sounded distinct and the vowels such as "a" could be pronounced /a/ or /ɑ/. Similarly the difference between é, è and ê (accents didn't exist at the time in the graphic system, they were invented later) would have been clearly audible. Then the diphtongues would have sounded differently as the oi for exemple was pronounced /we/ instead of /wa/. These are only a few things, but yes, it would have sounded different to the point that a contemporary French native speaker would have needed subtitles.