First off, I'm not a historian. Second, language development and historical linguistics questions are likely better suited to r/linguistics, maybe there are some historical linguists floating around here though.

However, the question needs to be reformulated or clarified. The Romance languages are separate languages from Latin - Latin is root but each Romance language is its own language with their own grammar, vocab, etc. An accent is simply a different way of pronouncing things; Brits, Americans, Australians, Canadians, Scots, etc. all speak English we just perceive each other as having accents because we pronounce aspects of the language differently. Dialects are a form of a language in a particular area; Quebecois French, South African English, Swiss German, Mexican Spanish, etc.

As the Roman empire spread out, the language of Latin outside of Rome into places like modern day France and Portugal began to fracture off into dialects called Vulgar Latin, not officially recognised by the state. As the Roman empire fell and, along with it, standardised Roman language, the Vulgar Latin continued to evolve and branch into what we now know as the romance languages. These days, speakers of romance languages can't understand written Latin or speakers of other romance languages. They are entirely different languages.

Nigel Vincent & Harris Martin: The Romance Languages

They're neither. A fairly early document that proves how distinct the different Romance languages was the Treaty of Verdun, which divided the Holy Roman Empire between the grandsons of Charlemagne and are written in Medieval Latin, Old French, and High German. The Latin and French translations are immensely different.

Even Old French is a very different language when compared to modern French. I'm not a linguist, but to my ears and eyes, I'd say they're definitely distinct languages.

"Dialects" of the same language are usually defined by linguists as being mutually intelligible with each other, even if they have grammatical differences. So, for example, American English and British English are both dialects of English. "Accents" are just different pronunciation patterns of the same language.

However, most pairs of romance languages, like French and Spanish, or Romanian and Italian, are not mutually intelligible, so they are neither accents nor dialects, but altogether separate languages.