I can answer the aspect of the question referring to the Anglo-Saxons, and can touch on the Celts and Vikings in relation to that. I did want to primarily focus on the Celts, but that would require explaining the differences between Gaelic, Brittonic and Pictish culture which is so complicated and so debated its not even worth it.

Firstly, I'm afraid that they were West Germanic rather than North Germanic, at least in terms of language. This might seem trivial, but it places the Anglo-Saxons in the contexts of the Frisians and the Franks (modern Dutch and French (and the Germans partially)) rather than of Scandinavia, and the Anglo-Saxons were to have far more diplomatic, cultural and trade links with the Continent than with Scandinavia, especially in the Carolingian period, after the Anglo-Saxon kingdoms had been more or less established. In regards to continental links during the Carolingian period, for a more diplomatic focus I'd recommend the Patrick Wormald's chapter on Offa, Alcuin and Charlemagne in Campbell's "The Anglo-Saxons" (Note: it was published in 1982 so some things that he states as fact have since been disputed) and for a more political history I'd recommend the chapter on Offa and Charlemagne in Wallace-Hadrill's "Early Germanic Kingship"

Secondly, you quite correctly draw the distinction between population and language in your question, and it is key that this distinction is recognised. While in 19th and early 20th century the general narrative was "the Anglo-Saxons arrive, beat up the Welsh and steal their land" the lack of Archaeological evidence for this has been used to dispute it, such as by Fleming in "Britain after Rome". Sometimes I feel like this argument gets a bit silly and goes too far: we must always be cautious using "negative evidence" (or "lack of evidence") to prove something. It seems likely that the plague of Justinian may have reached Britain, and we lack any archaeological evidence for damage caused by that; so it might also simply be the case that all the mass graves of the British who were slaughtered by the Anglo-Saxons have since vanished. Yet it seems to me that the arrival of the Anglo-Saxons was not one purely of conquest, but one of gradual cultural conversion by a new warrior class.

The withdrawal of the Roman army and power structure in the late 4th and early 5th centuries necessitated the development of some other form of government, which in this case was the various warlords or "tyrants" that Gildas refers to. While in the heavily militarised "highland" areas of Wales and Northern England, into which Roman style villa-based agriculture had not spread, a residual Brittonic (i.e. British Celtic) tribal structure persisted and eventually formed the much debated kingdoms of Rheged, Gododdin and (disputedly) Bryneich and so could effectively govern and repel Germanic and Gaelic raiding. The Hadrian's Wall garrisons by this point appear to have been isolated from the rest of the Roman World and were largely self-sufficient. It's not simply that they "went native", as Neil Faulkner points out that there was a gradual shift in the army in Britain to a more British identity, even as that same identity became more Roman. But it appears that the Roman local military forces in the North and West integrated into the subsequent tribal structures that formed, and gave them military strength. As such, the Anglo-Saxon kingdoms of Bernicia and Deira in the North found expansion inland from their coastal bases very difficult.

The same cannot be said of the very civilian and agrarian South, whose rulers governed in a Roman style but likely had a lack of troops to fight off invaders. Now the legend of Hengest, Horsa and Vortigern, in which Vortigern gives land to H&H in return for military service, after which he is betrayed by H&H and they conquer England, is much ridiculed, but I think that it might be a mythic version of a phenomenon during the 4th and 5th centuries in which local Romano-British rulers used Anglo-Saxons as soldiers to rule securely and to fight rival warlords. There is precedent for this in the Roman World in the extensive hiring of Germanic "Foederati", most infamously in the Visigoths. Bryan Ward-Perkins argues in "Why did the Anglo-Saxons not become more British?" that a fairly small number of Anglo-Saxons actually made the journey to Britain, but that they must have occupied a position in society that gave them immense influence, and allowed them to essentially convert the culture of their British neighbours. Additionally, it seems that Old English has a complete absense of any Celtic words, suggesting there was no intermingling and cultural exchange between the two groups. I don't believe that this was a relationship of subjugation, as English culture and language survived a Norman rule of centuries where British culture and language didnt survive in this regard.

Instead, it might be hypothesised that the Anglo-Saxon warrior-aristocracy seemed very attractive to the British, and the adoption of Anglo-Saxon customs may have been a tool of social advancement. Furthermore, the legal system which these emergent states had put in place by at least the 8th century, by which point they were thoroughly Germanic in nature, can only be described as apartheid; in terms of the "wergild" (the amount of money someone was worth in a legal sense) a British subject was worth half as much as an Anglo-Saxon subject according to Ine's Law Code in Wessex. I'd say that doubling your legal value is a pretty good incentive to abandon your British identity and Christian religion. As such, it seems to me that the Anglo-Saxons and their kingdoms actually had firmly British roots.

I'm aware this post has ballooned out of control, but I'd actually like to reserve the last paragraph for slagging off the Vikings. In my view, the impact of the Vikings on England is often overestimated. While the additions of their North Germanic language into our West Germanic language have made English fiendishly difficult to learn today, they are also often credited with the modern cultural divide between the North and South of England, as they established the "Danelaw" in the North. However, I previously argued that there was a distinct difference in the style of Roman governance between the North and South, and that the formation of Kingdoms differed in these two areas. So while the Vikings certainly influenced this divide, they hardly created it. The Vikings did not really influence ethnicity in Britain on their own, but they did have an immense impact upon the formation of other ethnicities, or "ethnogenesis". For example, Alfred's wars against the Vikings changed his kingdom from one of the "West Saxons" into one of the "Anglo-Saxons": the exterior pressure of the Vikings meant that the Anglo-Saxons were forced to change. Alfred's grandson completed (albeit temporarily) the reconquest of England, and by the time he did so it really was "England": in 927, Aethelstan changed his royal style from "King of the Anglo-Saxons" to "King of the English". The term "England" itself was also a result of the Vikings, as the first King of the "land of the English" rather than simply of "the English" was Cnut, a great King of Denmark who conquered England in the 11th century. And this impact was not limited to the Anglo-Saxons. In Woolf's "From Pictland to Alba", he makes it clear that before the arrival of the Vikings Scotland was divided between the Gaelic Kingdom of Dalriata and the Pictish kingdom of Pictavia: after the Vikings were driven out, these kingdoms had been replaced by Alba, which would become Scotland. So while the Vikings were themselves unimportant ethnically, they spurred "ethnogenesis" throughout Britain.