There is no one form of "Old English." We generally divide Old English into several different regional dialects: West Saxon, Anglian, Kentish, and Northumbrian. These dialects were formed not just because people who live in different regions tend to develop different speaking patterns, but also because the languages they originally spoke were slightly different. So speakers in one area might come from an Anglian speaking area on the continent, others from a Frisian area, others from a Saxon area. The difference between Anglian, Frisian, and Saxon were probably similar to the differences you might find between London speakers today and those in the west or north of England, or like the differnces in dialect between people in Ohio vs people in New York vs people in Alabama. More probably they were even more similar than the differences in those examples. In other words, they could easily understand each other, but they occasionally used a different word for the same object or a had slightly different accents. Children of people who spoke slightly different dialects would grow up speaking with a mixture of influences from their parents, and from the other children who came from similarly mixed homes.

This also means that the term "English" is not specifically referring to the Angles, but rather the mixture of all the Germanic peoples who came over from the continent. Although they did refers to specific kingdoms as "West Saxons" or "East Saxons," or "East Angles," in the tenth century after England was (sort of) unified by King Alfred, there was a tendency to refer to the whole country as Engla land (Land of the Angles). This perhaps came about from the influence of Bede who wrote his history of England in Latin in the early eighth century and in which he often uses the Latin word "angli" as a generalized term for the English to distinguish them from non-Germanic peoples. When this work was translated into Old English at the end of the ninth century, they translated "angli" as Angles. This translation is the first time that the island is referred to in Old English as "Engla land." It's a bit ironic since Alfred was actually king of the West Saxons.

Sources: A good place to read about the tribes who settled England is Fleming's recent book, "Britain After Rome." For the medieval English view of who settled England, read the beginning of Bede's "Ecclesiastical History of the English People."