@Thucydides:
The English as we know them are descended from a combination of the Celtic tribes inhabiting the British Isles before, during and after the Roman occupation, the Germanic tribes (Angles, Saxons, Jutes -- known more popularly as Anglo-Saxons today) who displaced those peoples in the first millennium CE, and Viking raiders who, to put it euphemistically, violently spread their genes all around the northern third or so of Europe.
The Celts are all over the place historically, including some tribes that made it to Spain, but that branch and the branch that entered Britain diverged before the emergence of the Celts in Britain. The Celts' likely origin is somewhere in between modern-day Germany and Italy. That said, they probably also intermingled with the locals at the time, who were probably of Iberian origin because the metalworking techniques employed by these people is of Iberian origin and these people would have predated the kinds of trading and commerce that would allow the technology to spread through different cultures to Britain (i.e. these people had to be from Iberia to know the Iberian technique).
And it's theorized, though I don't know how well accepted, that the Basque peoples you mentioned are the most direct descendants of Neanderthals that are homo sapiens.
...then again, by this logic, we're all partially Neanderthals, just as we're all partially apes, etc. =P