Or was it only a limited number of migrants (soldiers, traders, royalty?) which had an influence far beyond their numbers, with most people "Anglo-Saxon England" being Celts who had transitioned culturally until they were also Anglo-Saxons.
Since both groups were illiterate, what evidence do we have (archeological, genetic, linguistic)?
I recently saw two different BBC history programs that basically said the opposite things.
Oxford University and the Wellcome Trust conducted a wide scale genetics testing project (2004-2013) to investigate how closely we inhabitants of different parts of the British Isles are related to each other and how in turn we are related to other parts of Europe. The same size was 4,500 people and these subjects were chosen on the basis that all four of their biological grandparents were born in the same rough geographic location as them. This was to control for modern mass immigration.
In regards specifically to the Anglo-Saxon invasions they found that there had been a large influx, but this did not replace the pre-existing Romano-Celtic population, rather mixing occurred. That would suggest a high rate inter-marriage/breeding between the two communities. With the modern British population exhibiting both Romano-Celtic and Anglo-Saxon ancestry and concentrations corresponding to the areas of greatest settlement.
"the PoBI evidence points firmly to a large influx of Anglo-Saxon DNA but also the presence in modern descendants of a substantial amount of an ‘ancient British’ DNA which most closely matches the DNA of modern inhabitants of France and Ireland. This led the researchers to conclude that there had been an intermingling between the existing Romano-British population and the newcomer Anglo- Saxons, rather than a full-scale population wipe-out."
This seems to be a matter of debate. Some genetic studies have shown that there is an East-West continuum between East Anglia, where the genetic makeup of the population is predominantly Germanic, and Cornwall/Wales where the genetic makeup is Celtic, and in between you have some kind of mixture of the two. Other studies have shown that the vast majority of the British population are actually descended from the original inhabitants that came to Britain after the last ice age, and are genetically similar to the Basques.