My mother has always been a very keen proponent of the idea that the English population is indigenous, and has not been replaced several times though history via invasions. In fact this is now the established belief for at least the Roman and Norman invasions of the British Isles (54BC and 1066AD). The current belief is that a small number of invaders took up positions of power, but the majority of the population did not change, at least not genetically speaking.
The Saxon invasions of the dark ages however are more wildly disputed. A new study entitledY Chromosome Evidence for Anglo-Saxon Mass Migration claims to have found evidence that there was a large population change during the Saxon invasions.