This is actually a tricky and rather inconsistent thing, because though "Roman" survives in certain titles or references in law, Roman as some kind of cultural affiliation to the former western imperial state may not actually be trackable.

/u/GeorgiusFlorentius gives a fabulous answer here regarding the disappearance of Roman identity in southern Gaul that encapsulates many of the dilemmas. Namely, just because someone has a name that's Roman, just because someone calls themselves "Roman", doesn't mean they affiliate "Roman" with what you (the modern reader) affiliate the word with, namely someone who constructs their cultural identity as primarily belonging to the now-extinct but formerly western mediterranean spanning state lead by the emperors based out of Italia. What /u/GeorgiusFlorentius was pointing out, was that the people who were calling themselves "Romans" in southern Gaul, were probably constructing their identities locally as Franks, despite the Roman name, which by then was possibly just a geographic marker.

I would equate it to this. Assume you're a white anglo-saxon protestant living in New England. Do you consider yourself English? I assume you wouldn't, even though America used to be a part of the British Empire, much of its culture derives from England, and you yourself may be descended from colonists that came over from England. Despite living in New England and possibly being called a "New Englander", England itself has no real bearing on how you feel you construct your identity, despite the name.

This same situation regarding Roman identity was repeated throughout the western empire.

In Italy, especially the parts conquered by the Lombards, Chris Wickham points out the people there who were clearly descended from the Romans of Italy, ceased identifying themselves as such by the 8th century, as evidenced by the fact there are literally only half a dozen references to Romans being referred to in the record, out of all the surviving laws and charters. "Romans" were reserved for the Byzantines (evidenced by the survival of the province around Ravenna now called Romagna). By some time after Charlemagne, the rulers of the western kingdoms weren't even calling the Byzantines "Romans" either, they were calling them Greeks, especially once the west had set up their own rival "Roman Empire."

Also, even though Southern Italy was still under sporadic control of the Byzantines, it's clear that they constructed their identities as separate from that around Constantinople and Anatolia, despite being under "Roman authority". The Byzantine cities of the south of Italy had a strong independence streak, and were in constant flux and conflict with the lombard powers in the region, yet despite that, Wickham points out were culturally quite similar to each other. To them, recognition of political overlordship had no bearing on their own regional identities. Although with that said, I don't know that we can firmly say without a doubt that despite this regional identity, that they did not see themselves as "Roman citizens", so I have far more of a question mark as to when Roman identity would've ended here.

In Spain in the 7th century, Roger Collins points out you still have Roman names and people being called by senatorial or curial titles, but it's not clear if these are simply honorific titles rather than proper positions, as most of the Roman institutions had been co-opted by the Visigothic leadership. And of course, just because you accept a Roman title, doesn't mean you are a Roman, any more than Ronald Reagan being knighted by Queen Elizabeth II makes him British.

In Britain, Guy Halsall and A.S. Esmonde Cleary point out by the 5th century, the former urbanized Roman world of the province had completely imploded, and with political abandonment by the empire, the former denizens were forced to construct their own regional identities either from scratch (as the welsh) or from immigrants (the anglo-saxon migrants).

And of course, there's Roman Africa, the always forgotten bastard province of the western empire. Despite being under Vandal control for about 100 years, after their reconquest by Belisarius/Justinian, they were under eastern imperial rule for another 100, which was just enough time for the people there who had become Vandals to become Roman again. But then the Arab conquest came, and for reasons that are apparently still not quite known, there was a failure in the area to create an alternate subject Roman identity in the face of the Arab leadership one, and coupled with a complete reorientation of the trade routes due to the conquest, Roman Africa essentially broke down and factionalized into competing tribal identities at its base (which tended to merge with the existing Berbers in the region), with the Arab one at its top.

So back to your question, obviously there's a lot of grey here with regards to the disappearance of identity. But, as I'm sure you can guess, despite the grey, we can at least tell you where the boundaries of the grey are. Luckily, the conquests of the Caliphate and Charlemagne rolled in to clean sweep away a lot of older Roman identity, so those tend to be our firm end points.

tl;dr - When did Roman identity (as a cultural affiliation for the supra-regional empire) disappear? In the 400s for Britain, 500s-600s for Spain and France, 700s for North Africa, 700s for Northern Italy, 700s-1000s for Southern Italy.