They were from England!

A lot of people immigrated to England in the fifth century, including many from German and Scandinavia. People moved around inside England as well - one cemetery with which I'm especially familiar had more immigrants from Wales than Germany (based on a study of isotopes in the the bodies' teeth, which reveal where they spent their childhood). But these migrations were, all the evidence indicates, very unorganized and small-scale. The story told by Bede 300 years after the fact - that massive armies of Germans invaded in 450 and conquered half of Britain - is almost certainly false (see Halsall 2013 for a summary of the issues with story). There are a few other brief mentions of Saxon armies in England in other historical sources, but these texts don't tell us much, and all the archaeological evidence points toward largely peaceful migrations in the fifth century of farmers and small families.

These immigrants blended with the local population and started to rebuild the island's economy (which had collapsed in the late fourth century, probably as a result of the western Roman empire moving its capital east and diverting resources away from the Atlantic). Perhaps as early as the sixth century, but probably not until the seventh, several large kingdoms emerged which chose to emphasize the German - Saxon - part of their heritage. But it's important to emphasize that they were actually much more ethnically diverse; the choice to identify themselves as belonging to one particular tribe from a specific region of Germany does not, all evidence indicates, reflect a historical reality. It was a myth, used to unite smaller kingdoms under a unified identity.

It should go without saying that Anglo-Saxons didn't have any unifying physical characteristics, because their origins were multi-ethnic and diverse. People have tried to argue that they did; it was very common to measure the shapes of people's skulls in the 19th century (still is in some French archaeology...), but this has been conclusively demonstrated to not work. Härke argued that people buried with weapons in early medieval English cemeteries were Anglo-Saxons (because Scandinavians are tall, I guess?), but subsequent research by one of his own students has shown that these differences in height are a result of diet and social status (people with weapons were rich, ate better, and grew taller) and have nothing to do with German ancestry.

So really, we're not talking about a people group - we're talking about a political identity that was invented probably in the seventh century by kings who created an origin myth - based on a selective reading of their actual origins - which they used to unify their people.

Well, my brother belongs to a small distinct independent branch of Y DNA haplogroup I1, which is North Germanic. The ones in England have a MRCA around 500, and my brother's branch of them have a MRCA around 800, and they truly spread and diverged around the 13th century. They have a close relative in the southern Netherlands, and distant relatives in the upper Rhine, near the middle Elbe, and 5 miles from one of the trading centers on the south coast of the Baltic sea, which would be consistent with migration along the Elbe. (amber trade and later salt as well, and from the south, Mediterranean luxury groups and the Latin ancestor of the runic alphabet)

But, consensus of the people here is that my ancestors didn't participate in any particular migration to England. Mind, I can see possible ethnic diversity in that migration, but according to y'all, it never happened. My male line ancestors were picked up by UFOs, and dropped off in England, and probably along the Elbe and upper Rhine rivers as well.

Ain't science grand.