There's a little bit of a misconception built into this question, so I'll try to address that, and in doing so, I hope it also answers your question to a satisfactory degree.

The misconception is that there was some point in the past where everyone, from Portugual to Romania, were uniformly speaking something we'd all equally judge as "Latin", and then it diverged from that.

The reality is that at no point in time where these places all speaking the same thing, at least to a degree to which we could say there's a point of divergence.

I suppose the second misconception is that there's a time at which tings go from "the same" to "not the same" occurring after a considerable period of the former, rather than the reality which is that language is constantly undergoing changes, and no two consecutive generations speak identically. In other words, even if it were the case that at some point in time everyone in the Roman Empire spoke identically, and point at which they would start diverging would be "exactly one generation later".

What you actually see with Romance (and Sinitic, and every empire-supported language in history) is the effects of what are known in linguistics as substrata. You can think of a substratum is the influence a previous language has on the new language in a given area. If Russia conquered North America tomorrow and wiped out English in the continent, there would still be a North American flavour to the Russian spoken there, and it would be traceable back to North American varieties of English. In the same way, the English spoken in Minnesota has traceable influence from Swedish, or the English spoken in Singapore has traceable influences from Chinese.

So what we have in so-called Vulgar Latin (not a great name but one that a lot of people here are likely familiar with) is areas where Celtic influence was stronger (e.g. what's now France) and where that influence immediately coloured the Latin that was spoken the instant it was introduced and took hold.

We basically never had this one point where everyone spoke uniformly, but rather started out with what's called a dialect continuum, where town A is near town B, which is near town C and so on, and A and C can kinda understand each other, but town A people can't really understand people from town E. They all use the same written form, but that written form doesn't really reflect things happening in the spoken form of the language.

Then, to make matters worse, you have A conquering B, D conquering C and E and F, and H conquering G, so instead of A B C D E F G H you end up with A A D D D D G G, so people in area A who become Spanish speakers have some trouble with D who speak Catalonian or Occitan and they in turn struggle with G who speak French. In reality A and B were always different, just not so different, until B people started speaking A thus further separating their ability to understand people on the other side of the new border.

Of course there's one more matter to address, which is when X goes from being a dialect of something to being its own language. The answer to this question is: There's no scientific answer. It's entirely subjective. There is no single objective cross-linguistic way to determine, consistently and cross-varietally, what is a dialect and what is a related but separate language. These distinctions are 100% sociopolitical, and actually not something we really deal with in linguistics outside of sociolinguistic contexts. This notion of language vs dialect is basically not one that's made in linguistics, at least in most cases*, because it turns out it's pretty meaningless, in most cases*.

Hope that helps. Let me know if I can clear anything up.

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* When it's not meaningless is when you're studying language attitudes or planning/policy issues. But for people who do historical linguistics, the language/dialect dichotomy basically doesn't real.