Did the Danes only move to Denmark after the Anglo Saxons left or were they once both living there at the same time, but only the Anglo Saxons decided to migrate to Britain?
The Venerable Bede tells us in his history of the English People and Church that different tribes from continental Europe came to England to make their homes and that certain parts of the country were settled by certain tribes, the Angles, jutes, and Saxons, hence names like West Saxons, East Anglians, and so on. This is the view that has come down through history and is widely repeated in less academic writings on the subject. Only this isn't how it happened, and modern scholarship has harshly critiqued the old views on the subject of the Anglo-Saxon migration.
Robin Fleming talks about how the "Anglo-Saxon migration" was really a broader movement of North Sea adjacent peoples into Roman Britain. This included people from Denmark (Jutland), and Northern Germany (Saxony), but also people from Norway, Ireland, and Sweden. The idea of the Anglo-Saxons as a purely Germanic culture is misguided and not supported by the evidence that we have available through archaeology. She points to the blend of clothing and jewelry styles that emerged following "Anglo-Saxon" migration to Britain as evidence that these two cultures were assimilating into something difference from either that came before. She views this process as more or less a peaceful one. While they was some endemic violence inherent to the time period, she does not see evidence for the mass violence that is often assumed to have accompanied the Germanic migration into Britain.
These groups, both once in England and back within their own homelands however were not idle and static. Indeed this period of time sees particularly rapid changes in many different aspects of life. Based off of archaeological evidence this was the period that saw the first effective centralized rule in places like Denmark and Norway, emanating from seats of power and exerting influence over territory far from it. This is coupled with an increase in the availability of luxury trade goods as well as changes that can be traced in religious patterns. During this time we actually start to see the sudden ascendency of figures such as Odin in the material culture of Scandinavia, something that was not present during the earlier migration era, wherein the deities analogous to Tyr, the divine twins of Castor and Pollux, and Thor seem to have been far more common and widely worshiped. Other aspects of material culture changed as well. Burial patterns, jewelry designs, clothing styles, armor, weapons, and much more all went through a series of changes during this tumultuous period in history.
One thing that is paramount to remember is that these various tribal groups and "peoples" did not form coherent national identities that were set in stone and unchanging. This view of the angles, saxons, and jutes, forming one coherent polity and the British, Danes, and any other "ethnic group" another, oversimplifies the situation to an extreme degree and is an unfortunate holdover of the 19th Century. The period of the migrating peoples and the Early Middle Ages was not a time of Nation States, wherein the people of one ethnic group inhabited one distinct territory and were easily identifiable from their neighbors.
So the Saxons of Saxony and the Saxons who settled in Britannia might both speak the same language, worship the same gods, and so on, but they did not necessarily view themselves as the same "people" in an abstract sense of the word. This is in large part due to the rapid social and cultural changes that were happening in Scandinavia and Northern Europe at this time. This even ignores the changes that happened over time such as religious conversion/shift as well as changes in material circumstances.
Peter Heather argues that the identities of these groups were quite malleable in the social upheaval accompanying the end of the Western Roman Empire. Instead of kinship among these disparate groups of people, we should instead see loyalty between the armed retainers of a warlord/chieftain/insert your preferred noun here/ as the most paramount social identity. Status and position as an armed retained, a precursor to the later Huskarls and Housecarls, were much more important that subscribing to an identity of being "Saxon" "Anglish" or "Jutish" and that these ethnic labels were fluid and short lived as a rule.