It has been my understanding that the Angles, Saxons, and Jutes were invited in by the Britons to help in defense against the Scots and/or Picts after the Romans had decamped. As usual, I am open to correction.
That’s the story that Bede tells. If true, it is likely to be only part of the story. (Bede simplifies quite a bit and omits things that don’t fit the narrative he wants to impose on events.) Many were not invited. I’m not sure what Orosius and others have to say. If suitably ambitious (i.e., procrastinating from dissertation writing), I’ll look it up over the weekend.
The word invasion is not necessarily wrong in this context, but it can give the wrong impression (as can settlement). The Anglo-Saxon settlement of England was not like the Norman conquest, a short, decisive military campaign, but neither was it all hugs and bunnies. There were a series of wars/battles in addition to more peaceful settlement and trade. (The Viking invasion/settlement of England was much the same.) And the Anglo-Saxons did replace the British socio-political hierarchy in what is now England.
The mostly non-Romanised Britons living in the west and north of Britain were largely unaffected by the Anglo-Saxon settlement.
That’s flat out wrong. They weren’t conquered, at least not right away, but unaffected isn’t a word I’d use. By the time the Normans came on the scene, the Anglo-Saxons were pretty much in control of Cornwall. The Normans would then take on Wales and Scotland, a process of subjugation that would take several more centuries. (And some would argue has never been complete.)