Well, people didn't forget their Roman past very quickly. All of the post-Roman kingdoms in the west were ruled by Romanised elites except for Britannia and they all preserved Roman institutions and learning to the best of their ability. We might call the Frankish kingdom Frankish because of the ethnicity of their leaders, but the laws they enacted, the manuscripts they copied and the titles they assumed all had Roman roots. These elites were also always a minority of the population, so the majority of the population remained the same as when they were ruled by the Roman emperor directly. There were cultural changes, but that took a very time to occur and many of these regions had been under 'barbarian' rule already well before 476, so for the general population very little had changed. As for the east, well, the Eastern Roman Empire lasted for another thousand years. Admittedly North Africa, Egypt and the Levant were all lost to the Arabs in the seventh century, but Christian culture continued to thrive there for centuries afterwards.

So why wasn't there just one nation? Political division is the easiest answer, since if one powerful leader wanted to rule a piece of territory, there was little to stop him if he was undefeated. Moreover, there plenty of attempts to recreate a larger state, such as King Theodoric the Great's attempt to bully the western kingdoms into joining his informal Ostrogothic empire, or the reconquest of Italy, North Africa and parts of Spain by the eastern emperor Justinian, both of which happened in the sixth century. The rise of a Frankish empire in the eighth and ninth centuries can be seen in the same way. The ghost of Rome never really left Europe, for if it was so easily lost, why would people as far ranging as the Ottomans and the Russians claim it as part of their heritage as well?

Why soon after the fall of rome there was already french, british etc

As /u/soprescribeme mentioned, one reason is that the groups who formed the French / British nations were non-Roman tribes from outside the empire. Linguistically France derives from the Franks, England from the Anglo-Saxons. Of course many of the people living in France / Britain had been ruled by the Romans, but the dominant groups came from elsewhere.