Your first sentence, though commonly held, is thought to be debunked by genetic evidence. The modern English, Irish, Welsh and Scottish are genetically very similar, almost to the point of being identical. The idea of the Angles, Saxons and Jutes immigrating to Britain in vast numbers and displacing the native Celts* seems to be a myth. The real truth, according to the genetics, seems to be a cultural and military subjugation, more on the lines of the Roman conquest of Britain. The Anglo-Saxon invasion contributed less than 5% of modern English DNA.

*It is worth noting that the Celts were not the original inhabitants of Britain either. There were people in Britain before the Celts came over, though sources disagree to the culture and language of this earlier population (but most likely Iberian/Basque). Again, the Celtic 'invasion' (for want of a better word) resulted in less than 5% of the genetic imprint being changed.

In fact, genetic studies show there is very little difference between any of the modern Western Europeans; English, Scottish, Welsh, Irish, Belgian, Dutch, Germans, French, Spanish etc, show little genetic difference. The explanation is that groups and tribes continually migrated, invaded and mixed with each other. The idea that Britain was wholly 100% Celtic and different to mainland Europe before the invasion of the Romans seems to be false. The fact that there are very few Celtic placenames in England seems to confirm this - "Celtic" England may not have been speaking a Celtic language before the Romans came.

http://www.prospectmagazine.co.uk/magazine/mythsofbritishancestry/ http://www.sott.net/article/263587-DNA-shows-Irish-people-have-more-complex-origins-than-previously-thought

sub Q: Any evidence of the impact of the romans on the genetic makeup? (everyone bangs on about the Anglo-saxons).