While u/kuzyn123 gave a correct and very concise answer, I‘ll try to expand on it.

We can’t always say for sure how non-writers perceived the events of their time. Most of our written sources either come from Roman perspective, filled with stereotypes, simplifications, political narratives or simply a Roman-only perspective. Meanwhile the first Germanic perspectives either come from religious authors (like the Wulfila/ Gothic bible) or from chronicles written during the 6th and 7th century which have a strong Roman influence, are heavily mythologised and also have strong political narratives to legitimise the very young and newly established Christian-Germanic kingdoms.

Due to that we don’t exactly know what the „Fall of the Western Empire“ looked like from a Saxon point of view and what priorities they had. Even today the term „Fall of the Roman Empire“ is very debated and we only generally use the date 476 AD because the last Western Roman emperor was deposed by Odoacer (but it could also be 480 AD because of the murder of Julius Nepos or 486 with the fall of the Roman rump state of Soissons under Syagrius).

The problem with the „fall of the empire“ narrative is our involuntary association with power vacuums, chaos etc. and then we ask ourselves if this was even noticed by forces outside of the empire. But in reality the Ostrogothic kingdom of Italy continued the Roman administration from 476 onwards, the Merovingians emerged from Gallo-Roman troops and defeated the „Usurper“ Syagrius, thus legitimising their claim on the land (Frankish tribes had been guarding the limes for nearly a century at that point), the romanised Visigoths had been imitating Roman coinage and administrative practices since their settlement in Aquitaine in 418 and despite their ongoing conflicts with the Western Empire they still fought against Roman enemies like the Suebi or the Huns.

Let’s say you’re a Saxon chief in 500 AD and, say you’re 40 years old at that point so you were old enough to „witness“ the deposing of Romulus Augustulus, however you probably didn’t care much about that. Puppet rulers in Italy (beyond the alps, so quite far away) had been installed and deposed on a yearly basis the years before and now a new leader rules that area. Much more concerning would be the establishment of the Frankish kingdom to the west under the now Christian Gallo-Roman rulership of Chlodwig.

Meanwhile, in Britannia, the „rulership in Italy“ had abandoned the island 90 years ago. Since then, your ancestors (and that of your allies and tribesmen from the same region) had been serving as Roman mercenaries (Foederati) to defend the local Romano-Celtic population. The leaders of the Romano-Gauls had conquered Angers and kicked out the Saxons in 463, three years after your birth, so if you associate the Franks with the Romans at that point, (especially after their Christianisation) then they looked pretty alive and not-so-falling to you in 500 AD. At the same time (the exact time is heavily debated) more members of your tribe, and those of your allies have been sailing to the island where your ancestors had served as mercenaries and now a fight has broke out between them and the local population.

I focused on the Saxon perspective now, so keep in mind that every perspective is different, even in Saxon tribes themselves and even more when looking at different larger tribes and coalitions (Franks, Aquarian’s, Thuringians, Burgundian etc)

My main sources (I‘m broadly working on a paper about Germanic tribes with focus on the goths so I had to quickly read up on Saxons). Especially Walter Pohl (my main source for the goths) also mentions the Saxons and their migration on several occasions in a broader context but also Peter Heather and Roland Prien