From what I understand, the people of pre-Saxon England/Lowland Scotland would have spoken a Brittonic language related to the ancestors of modern Welsh/Cornish/Breton. The Anglo-Saxons supposedly did not wipe out the native populations of Britain upon their arrival, but simply took positions of power. Consequently, the Anglo-Saxon language became the language of prestige and the native Britons simply adopted it, displacing the Brittonic languages to such an extent that there is very little evidence of Celtic influence on modern English, aside from place and river names etc.
However, Old Norse and Norman-French did not have the same effect on English. Despite holding significant positions of influence within the English state, the Danes and Normans only influenced the language, they didn't entirely displace it, leaving almost no trace of the original language. Why was there such a difference in the level of effect on the native language that these invasions had? Is there any other instance of a populace almost completely abandoning their native language to adopt the language of their new overlords? From what I understand, even in former colonies where large portions of the native population now speak European languages, the European language is often significantly influenced by native languages.
Let me first say that I'm still a student but I've studied the history of English for some time now. This question is a little politically charged and it has many layers which I'll try to tease apart.
For reference, here are the periods of the history of the English language:
Question 1: What was the relationship of Old English and Celtic languages?
As you say, English likely became the language of prestige (superstratum). To survive, it was beneficial for speakers of the non-prestige (substratum) language to adopt the new language. This is known as diglossia and takes different shapes. What's typical of this situation is that the substratum language (here: a Celtic language) adopts a large number of features from the superstratum language (here: English), but this is not the case the other way round.
Question 2: To what extent did the Anglo-Saxons wipe out the native populations of Britain and take positions of power upon their arrival?
The dominant narrative, until very recently, was that Celts were mass-exterminated and forced to migrate. There is a new narrative called the Celtic Hypothesis that says the Celtic substratum consisted of around 80% of the population and the Germanic superstratum of only around 20% (and a good layer of Latin over everything). Celtic speakers start learning OE gradually, in terms of time and territory, passing it on to children as mother tongue. The Celtic languages transfer a whole bunch of things into the new language, like sentence structure (syntax) and accents (phonology) (things you would expect to transfer when you're learning a new language), but not word items (lexis) themselves (as you probably wouldn't when speaking a second language).
Now, as far as I understand, there are elements found in ME that would support this hypothesis. There is also an argument for some minor evidence in OE. Another piece of support is that it seems that a lot of people on the British Isles today have a lot of Celtic genes. However, there are serious problems with the Celtic Hypothesis. The clearest candidates for Celtic influence appear only in late ME, not in OE. One might assume that a conservative dialect of OE, West Saxon, has prevented Celtic influences from appearing in writing, but this is quite an assumption to make. For the rest of the answer, I will sort of subscribe to the "old dominant narrative." As all this stuff happened hundreds of years ago, it's very hard to know for sure. It may well be possible that English developed supposedly Celtic structures in parallel to those languages.
Question 3: Why did Old Norse not have the same effect on English as English on Gaelic languages?
In the sense that were was an aspect of diglossia, it did have the same effect. Old Norse influenced OE to a great extent. However, Viking attacks happened during a much shorter time period and they took less land. The first viking age during which parts of the British isles were raided and later settled lasted from around 787 AD to 880 AD, after which the vikings were relatively peaceful. They coexisted with the Anglo-Saxons and made agreements that their law would apply to the lands they conquered. Later, the Anglo-Saxons conquer back much of the land in the 910s and 20s AD and the vikings retaliate. They defeat the English and start raiding again and the Vikings only actually rule England from 1013/6 until 1066. So their time window for influence on the whole area of England is relatively small and this may be why English wasn't ass affected as Gaelic languages.
Question 4: Why did Norman French not have the same effect on English as English on Gaelic languages?
In 1066, the Norman Conquest swept across the British isles. This time, Norman French settlers took over England and established French as official language for law and government (two words borrowed from French! I wonder why). French affected English greatly, probably more than any other language. Written English went extinct and resurfaced 200 years later as ME. During the Hundred Years' War, in 1420, English again became the new official language for law and government. After that, the English and French monarchies remained separate and William Caxton introduces England to the printing press, which cements written English and helps propagate a new standard.
This took way longer than I anticipated, so I will have to capitulate after this. The source for most of this are my lecture notes. A good book where you can read most of this is
Brinton, L. J., & Arnovick, L. K. 2017. The English language: A linguistic history. Oxford University Press, Canada. (Third Edition)
Also, here's a useful diagram guessing what languages were used how much and when from
Manfred Görlach. 2001. “A history of text types: A componential analysis.” In: Towards a History of English as a History of Genres. Ed. Hans-Jürgen Diller and Manfred Görlach. Anglistische Forschungen 298. Heidelberg: C. Winter, 47-88.
My professor also recommends this:
Malcolm Falkus & John Gillingham. 1981. Historical Atlas of Britain. London: Book Club Associates.
David Crystal. 2004. The Stories of English. Woodstock and New York: The Overlook Press. (Chapter 5 and 6.)
Final disclaimer that I'm not 100% confident in the correctness of all of this. My professors might rip their hair out if they saw this. I'm especially not 100% sure what exactly is part of the Celtic Hypothesis.
The other commenter gave a very solid answer with regards to the language argument. However, I want to correct some misconceptions in this question about the nature of the Anglo-Saxon arrival in Britain, and how it was different from the Vikings and Normans.
Around twenty years ago, what you've written was indeed a commonly-held viewpoint, mainly amongst archaeologists (i.e. the fashions and language changed but the people didn't). But now we're accumulating a lot more evidence, particularly genetic and isotopic evidence from ancient grave sites, and it turns out that a lot of the people buried there were indeed "genuine Anglo-Saxons." And these Anglo-Saxons were not elite warriors by any means. Based on grave goods, a lot of them looked quite poor, and many were women. It may not have been a genocidal "invasion" as the traditional view of the process suggests, though violence cannot be ruled out, but in general the early Anglo-Saxon settlers were encountering a fairly demoralized society that had already experienced significant population decline. Recently, scholars have suggested a process of chain migration, which is to some extent implied in Bede's Historia Ecclesiastica. Either way, it's clear to most nowadays that, at least in southern and eastern England, we are dealing with a large-scale folk migration rather than an elite replacement. However, as the Anglo-Saxons expanded north and west it's possible that they could have developed a sort of social apartheid which would have encouraged Britons to emigrate or adopt their culture. Thus, you have the spread of the English language.
Elites rarely replace a language, though they do often influence it. There are exceptions--Latin across the Roman Empire for example--but these are typically associated with an extremely dominant, resource-rich society that rules over a large area. Old English was nothing like this.