If Anglo-Saxons came from Schleswig-Holstein, would not then the English be actually German?
No. “Anglo-Saxon” as a name already shows the mixing that was going on what would become “England”. The Angles and the Saxons were separate tribes who came over from the European mainland around the same time, along with the Jutes. The Angles came first, at the invitation of Vortigern, in 449. The Jutes were from Jutland, in modern Denmark, along the Frisian coast. The Angles were from Schleswig in what is modern Germany. The Saxons, judging from the writings of Tacitus, Ptomoly & Bede, moved around quite a bit within the territory that we now call Germany and Holland. When members of these tribes came to the island of of Britian, there did not and were not considered “German” at home. Likewise, the fact that large numbers of the Belgae tribe migrated to Britain does not make the English actually Belgian. And, as other have noted, “English” is also influenced by the pre-celtic peoples (probably related to the Basque), the Picts, all the various p-Celtic and q-Celtic tribes, the Romans,
