The idea that the Celts were eradicated—culturally, linguistically and genetically—by invading Angles and Saxons derives from the idea of a previously uniformly Celtic English landscape. But the presence in Roman England of some Celtic personal and place-names doesn't mean that all ancient Britons were Celts or Celtic-speaking.
The genocidal view was generated, like the Celtic myth, by historians and archaeologists over the last 200 years. With the swing in academic fashion against "migrationism" (seeing the spread of cultural influence as dependent on significant migrations) over the past couple of decades, archaeologists are now downplaying this story, although it remains a strong underlying perspective in history books.
Some geneticists still cling to the genocide story. Research by several genetics teams associated with University College London has concentrated in recent years on proving the wipeout view on the basis of similarities of male Y chromosome gene group frequency between Frisia/north Germany and England. One of the London groups attracted press attention in July by claiming that the close similarities were the result of genocide followed by a social-sexual apartheid that enhanced Anglo-Saxon reproductive success over Celtic.
The problem is that the English resemble in this way all the other countries of northwest Europe as well as the Frisians and Germans. Using the same method (principal components analysis, see note below), I have found greater similarities of this kind between the southern English and Belgians than the supposedly Anglo-Saxon homelands at the base of the Danish peninsula. These different regions could not all have been waiting their turn to commit genocide on the former Celtic population of England. The most likely reason for the genetic similarities between these neighbouring countries and England is that they all had similar prehistoric settlement histories.
The essay makes for fascinating reading. Oppenheimer also has a book forthcoming on the subject.
(Thanks, Alex Hague, for the link)