The important thing to consider about the 'fall' of the Western Empire is that most often, despite the doom-laden writings of writers like Procopius or Gildas, the Empire didn't fall to anybody, it just kind of fell apart. So people weren't enslaved or killed because there weren't invaders to kill or enslave them. Obviously this happened to some extent, especially in border territories, but mostly what happened was just the reversion of authority to local pre-Roman or tribal structures, hybridised with surviving Roman governmental structures and the Church.

In England, for example, the Romano-British authorities tried to continue as best as they could without the logistical, economic and military support of the wider Empire, but as Gildas shows, they simply weren't able to hold things together without it, but things disintegrated slowly rather than overnight. While the Anglo-Saxons did raid and plunder before their settlement, archaeological and dna evidence suggests that the majority of the population in England remained ethnically Romano-British rather than Germanic. Indeed, only approximately a third of the population was likely Anglo-Saxon, concentrated more in the elites of society. Some nominally Anglo-Saxon Kingdoms may have remained Brythonic however - The Magonsæte of modern Herefordshire for example appear to have been ruled by a Romano-British rather than Germanic elite well into the ninth or tenth century.

The answer to this question changes greatly from time to time, and from place to place.

A few months ago I wrote this large overview of what happened in the various parts of the empire, and how life changed for its inhabitants. There are also many follow-up questions I answer.

But a much more in-depth answer can be written about any of the different parts of the empire. Although sometimes, such an in-depth answer can't really be written yet, as we're heavily dependent on archaeology and some areas just have not seen as much work done as others.

u/shlin28 also wrote one of my favourite posts on how the city of Rome itself changed in the centuries after the fall of the central empire, here Definitely worth checking out.