Before the Romans left, late 300's early 400's, the population was mostly Celts, English, Welsh, Scots and Irish. The first invaders were the Anglo-Saxons and I think that was a gathering mass migration from West Germany down to modern day Holland, not just leaders like the Normans. England gets it's name from the Angles from Southern Denmark and NW Germany, courtesy Alfred the Great.
Subsequently, there were incursions from the Danes and Norwegians (Vikings). Went on up to c1000. By this time, population was mixed Celts, Saxons, Danes and Norwegians. In the North you get place names ending in by or Thorpe added, the area of the Danelaw.
The Vikings founded Dublin but were pushed out by the Irish Celts and resettled in Cumbria. Melvyn Bragg who hosts 'In our Time' Thursday mornings on Radio 4 proclaimed his pride in being a Viking within the last year. Alfred the Great did not get that far north.