This is an excellent question which is very much dependent on the geographical area in question. To begin on a basic level, there was no consensus between the newly created barbarian polities as to reunify as fundamentally there was nothing left to unify. To see the successor states as simply offshoots of the Roman state is fundamentally flawed; the Vandals, Burgundians, Visigoths, Suevi, Alans etc.were distinctly different groups with their own laws, customs and practices. There was little desire from the barbarian kings in the West to unify as the West was now made up of competing, independent polities with their own interests. The West was now without the unifying concept of 'Romanness' within the ruling elites which had been present under the Western Empire. Some of the new barbarian kings in the later 5th century, such as Clovis ruler of the Frank's, had not been members of the Roman political system like earlier barbarian leaders such as the Visigoth Alaric and had such held no desire to maintain its political systems or placed any value on the concept of maintaining any form of imperial unity. In the chaos of the later 5th century it would have been extraordinarily difficult militarily and practically to conquer all the territory of the former Western Empire; there is a reason that it took the Frank's until the middle of the 6th century to conquer what had been Roman Gaul and Germania from various other tribes and until 800 for Charlemagne to unite most of what had been the territory of the empire in the West.

We cannot forget that to the new rulers of what had once been the Western Empire the Roman Empire was still alive and well in the East. When Odoacer deposed the 'last' Western Emperor Romulus Augustus in 476, he did not take the title of Emperor as he could have done but rather that of King. Although he held real power in Italy, he paid lip service to Zeno (emperor in the East) and modelled himself as ruling in Zeno's name. To declare yourself as the new 'Roman Emperor' in the West and claim the imperial legacy would have been a serious challenge to the Eastern Roman Emperor and empire as a whole which was a serious risk both politically and militarily. Even when Charlemagne was crowned Holy Roman Emperor in 800, the ruler of the Byzantine Empire Irene of Athens was furious as to her she was the legitimate ruler of the true Roman Empire.

The experience of the citizens of the former Western Empire as to the fall of the empire varies due to geography. In Britain, for example, the Roman structure of everyday life collapsed throughout the former Roman territories and this would have been extremely noticeable. In Italy, by stark contrast, life went on much as it had done under the Empire and very little changed for the ordinary Roman. There was little to miss about the imperial era for a simple peasant farmer, for example, who would live life very much as he had done under the Western Empire. Guy Halsall argues that for most Romans in the West outside of the educated elites there was no catastrophic realisation that the empire had collapsed; they still considered themselves to be 'Roman' and other than slight changes in the way they were taxed and land was distributed life would have been remarkable similar. The ordinary Roman in areas such as Italy and southern Gaul would most likely not even have noticed the empire had disappeared for sometime after 476.

The level of continuity from the imperial era in some of the successor states was actually surprisingly high. Rule in Ostrogothic Italy, for example, was surprisingly similar to that of the preceding emperors. Theoderic, king of Italy between 493 and 526, minted coins with his own image in the style of an Emperor, wore the imperial purple garb and held annual games and celebrations for the populations of various cities in Italy. The system of government of the late empire, despite being notoriously corrupt, was more effective and easier to maintain than the methods of barbarian groups and as such some kings like Theodoric maintained many of the systems of Roman government. There was a serious amount of prestige in drawing implicit links between yourself and the image of a warrior emperor for a barbarian kings and the Roman legacy was seen as one which could give legitimacy to kings ruling in the West.

To summarise, to be blunt there was no consensus in the West after the collapse of the Western Empire as there was no longer any unified form of political governance. There was no real chance of one state such as Visigothic Spain or Ostrogothic Italy from taking all the territory of the Western Empire back and it would have been foolish to even attempt it in the later 5th century. Beyond this, there was little desire to resurrect the 'Western Empire' as barbarian kings either didn't believe in it (understandably as they were not Roman and had not been involved in Roman politics) or continued to use Roman administrative systems and symbols to gain prestige and for ease rather than out of any serious desire to reunite the empire.

I can go into much greater detail about the experience of the Roman elites in Roman Gaul during the mid to late 5th century if you wish. Gaul is perhaps the perfect example to view the Roman experience of the decline of imperial authority and the different reactions to it.