Well let's look at a case study in the opposite direction and then circle back.

Britain. Britain was a more or less fully Romanized province of the Empire with its own Romance speaking population, form of Roman urban life, and unity within the broader Roman economic and political world. Then it wasn't. The why isn't super important, the tl;dr is "terrible economic crisis". This resulted in quite a few important knock on effects, namely the collapse of urban life in Britain, the destructing of Britain as a politically cohesive whole, and the removal of Britain from the broader Roman economic world.

In the wake of this economic collapse and the coming political collapse of Roman power in Britain, Roman identity lose a great deal of its cultural cachet in Britain All of a sudden it became a lot less important to look and sound like the Romans than it did the newcomers. Assimilating to Anglo-Saxon society was the new way of career advancement and opportunity, not embracing Roman identity. Now it might behoove the native British to look and sound like a Germanic person. Now through in all the other things humans do, they intermarry, they live next to each other, and so on. This eventually created a new cultural identity that was not the sole production of the wholesale slaughter of the native British by the Angles and Saxons, but by the assimilation of the Germanic newcomes and the native British into a new culture, that while much more obviously Germanic, did retain aspects of Roman life. Robin Fleming breaks down this entire process in her Britain after Rome and while there is some disagreement about the scale of violence in this transitional period, I for one think she down plays it far too much, this central narrative of assimilation is currently quite dominant.

So how is this different from the experience of Iberia, Gaul, Italy, and so on? Well for one, Roman cultural cachet never totally lost its luster, and to say it did in Britain would be a mistake as well (both the surviving British polities and later Anglo-Saxon polities would later ape Romanitas, or the quality of being seen as Roman) We should also keep in mind that this is perhaps a question of numbers as well. While it is inconclusive and controversial among historians, some DNA evidence has suggested that the modern English on the east coast have a higher level of genetic material from Scandinavia and Northern Germany. These findings are not conclusive and are still controversial and may be problematized by later movements of peoples. Yet if this number is anywhere near accurate, we are seeing a truly mass migration into England, nearly 1/3, probably closer to 1/4 or even 1/5th of the populating being newcomers. This far outstrips any sort of estimate for the number of newcomers into Italy, Gaul, or Iberia, and so on. While these groups may have attempted to delineate themselves as a separate ethnicity as we see in laws such as the Salic Law, at first the newcomers would have absolutely paled in number to the established population.

Peter Heather argues that the groups such as the Visigoths and Ostrogoths were not even ethnic groups originally, but essentially armies that eventually coalesced around a shared identity even if the actual constituent members of the group were far flung originally. But armies, even very large ones, could not really be as thick on the ground in places like Gaul and Italy as they could in Britain.

Furthermore in Gaul, Italy, and Iberia Romanitas never really lost sway. There might have been some differences in dress, one certainly can imagine Romans gossiping about how the uppity social climber is now sporting a mustache and trousers instead of keeping to good Roman fashion, but by and large Roman identity managed to hang on and retain some cultural status and importance on a level it did not in other areas.

This extended to language as well. Language assimilation was not a quick process but it nonetheless happened. Indeed there are, few admittedly, loan words into French and Spanish from Frankish and Visigothic, but they do exist. We really shouldn't imagine that overnight the Franks and Goths woke up and decided to start speaking the local variant of Latin, it was a slow process taking a few centuries to really take hold.

Certainly in much of these areas romance would have been the language of the majority of people and even of a relatively large number of aristocrats and other notables. Now we shouldn't envision the newcomers shedding their language in an effort at greater understanding or in the interests of clarity, it was a far slower process. Some individuals may have elected to learn the language and teach their children, others would have learned it from nurses and mothers, but there was still a great deal of importance placed on maintaining seperate identity as a Frank for example, just see the Salic Law on how laws applied differently to Franks vs Romans.

So in these areas you've mentioned the Germanic newcomers were a tiny fraction of the population, and despite efforts to strictly mark off Romans vs Franks for example in practice these sorts of identities were fluid and relied a great deal on societal expectations as well as a slow process of assimilation.