The core premise of your question, that the arrival of the Anglo-Saxons caused a significant displacement of people, is flawed. It is highly unlikely that there was a mass migration of the pre-existing inhabitants following the arrival of the Anglo-Saxons for a number of reasons, such as the actual difficulty for all but the wealthy members of society to actually up and leave. The regions that the Britons were traditionally thought to migrate to (Wales, Cornwall, Scotland, Brittany) were already inhabited, and are not that open and fertile to be able to accommodate the arrival of thousands of people looking to live there, particularly Wales, which is a mountainous country. Rather, if there was a migration, it was the elite who left, travelling to relatives or other elites they had ties to who could establish them in their new home.

There are hints in the literary record that the Anglo-Saxons actually had to accommodate a largely Briton population after they came across. For example, in Wessex, the names of early rulers of the house of Cerdic have names that appear almost as if they were a mix of Germanic and British Celtic. This book is a good overview of the evidence.

Moreover, genetic evidence of remains is unreliable at best. The sample size is always limited and the things we can actually extrapolate from such findings are restricted at best, such as social standing.

I recommend this answer from u/alriclofgar for a better answer.