Say you are in modern-day Great Britain, all your life, you've lived under the Roman Empire and one day, you're told that the Roman Empire was no more. What would the general thought process be in the wake of such collapse around the Roman Empire?
Thing is, nobody told anyone the Empire was no more in the Vth century. The idea the empire was undergoing a severe crisis in its western provinces was certainly there, and in its more peripheral regions (Britain, Illyricum, northern Gaul) the imperial authority collapse was certainly obvious : people were aware the political situation did change.
But it did change, for them, as it often did so far : Romans were already accepting that the Empire had to reinforce its state appartus bypassing provincial administration and refinforcing political pressure, that it had to be regularily split over the IVth and Vth century into two main halves and futher down between more or less equal co-emperors or junior lieutnants, and that Barbarian communities could get important territorial autonomy as foederati to the point being essentially autonomous.
In some aspects, the end of the line of western emperors was more of a mercy killing of an ineffectual and irrelevant rule of various usurpers that restored the unicity of the Roman Empire at the benefits of the emperors residing at Constantinople : barbarian kings formally acknowledged themselves as these emperors' lieutnants. For instance, Sigismund sent this letter to Anastasius, stressing that
[Burgundian] people is your people; I obey you as I command them : I am a king for my kin but I am but your soldier
Up to the the VIth century, Barbarian coinage was made in the name of the emperors, Barbarian kings displayed Roman regalia and honors, and the alliance with Constantinople was determining into allowing them to rise up (such as Theodoric replacing Odoacer and assuming sort of vice-royalty for the former provinces with imperial support, or Childeric and Clovis alliance with the Empire allowing them to warlord their way into domining most of Gaul)
Now, arguably, this wasn't the case in the aformentioned most peripheral regions : while Roman institutions survived the political collapse of the Roman state in the West in Italy, Africa, Spain and (most of) Gaul, it didnt in Britain or Illyricum : in the former region, particularily, the state apparatus was essentially gone and late imperial elites adapting themselves to a simpler hierarchy based on clientelism. But even in Britain, these elites still maintained reference to their romanitas by maintaining trade contacts with the Empire : ships from the eastern Mediterranean basin still went to southern and western Britain to trade goods associated with Romanity, contacts with Barbarian kings stressing a certain form of political continuity with Constantinople were important into the political frames of these peripheral regions in the same way. The Empire certainly recedeed, it certainly changed and not necessarily for the best (altough there weren't only negative effects, namely a lesser fiscal pressure) they might not be part of it due to the lack of legitimized overlords, but it was there, it existed and could be contacted for trade, politics and identitary purposes.
At this point, you'd probably argue that it's well and good, but effectively they were part of new polities that if considered themselves part of a Roman and/or Christian commonwealth were virtually independent and that Romans eventually had to fight in the VIth century assessing their hegemony in Central Mediterranean basin. You'd be entierely right : the Empire still existed but lost direct authority over its western provinces where Barbarian kings essentially took its mantle, the fiction of them being subservient to the Empire being mostly a political and symbolic device of power (which, to be fair, isn't vastly different from what happened in Vth century western Romania with Barbarian and Roman warlords). But this fiction wasn't born out of cynicism or play-pretend : it was the continuity of a political make-up where Barbarian legitimacy depended from being in relation with the Empire.
From the Vth to the VIth centuries, this fiction co-existed with independent politics of suceeding kingdoms, possibly as well with the emerging polities of post-Imperial Britain or Illyricum, in no small parts due to Imperial diplomatic policies of managing subsides, honors, gifts and trade to their courts (and thus keeping a line and display of wealth and power to them). Gregorius of Tours, living in the half of VIth century Gaul still accounted for 2/3 of its population being Roman.
What, gradually, changed is that Roman and Barbarian identities were importantly political in their definition : who followed a Barbarian king was considered a Barbarian, who directly served the Roman state was a Roman. Significant identity features were stressed to preserve the distinction as both were increasingly culturally similar, such as clothing display, religious upbrining between Homeism and Niceism, etc. but when the Barbarian king not only ruled over its own people, but as well on an imperial region forming its own administration and palatial network in replacement of the imperial one... Things got more difficult and more contextual : being a Roman was a matter of self-identity as surviving military and civil administration served a romanized, but Barbarian state and eventually depended from their own social autonomy and connection to the eastern Empire.
By the late VIth century, Imperial impopularity in the West (for reasons hard to really determine, but that might probably have involved religious devellopment, Imperial takeover of Italy, lesser trade and connections due to the wars with Sassanians, epidemics, etc.) did severed several mental links, and Roman identity tended to disappear except from Imperial Italy and southern Gaul. As provincial and Barbarian elites and populations merged (and especially in regions such as Britain where new Barbarian polities emerged from an indigenous development favouring the culture of newcomers), as the immediate reasons for calling yourself a Roman were gone except in regions with strong regional tensions, the idea that the west was still, somehow, part of the Empire or an imperial commonwealth was gone.
Now, this wasn't just a matter of immediate reasons, as the mass of Roman or early medieval texts allowed some momentum for a lasting identity, but these weren't accessible to most, and the new political situation was nevertheless a strong factor into the making of new identities. Furthermore, what became central to these new identities were not only political but importantly religious in nature : early medieval authors as Gregorius, Paul, Bede, did mention political or ethnic identities, but also stressed the historical and social importance of Christianity into the making of people, something that simply wasn't there into Roman historiography whom Christianisation was relevant, but more of a crowning achievement than a re-birth.
Eventually, while we can point how the collapse of the Empire in western and central provinces was indeed an important change and a thresold into the social and political make-up of these regions (not only in matters of identity, but as well economics or culture), it wasn't sudden or felt as a radical transformation by contemporarians : rather it was a set of gradual changes from, specifically, a Roman identity and the idea being part of the Empire to new identities over the time until, at some point, the worldview had changed, depending on the region, the immediate social and political context, and other regional factors.