Well there are a few issue to get at here. I can really only speak to the historical Saxony and the Saxons who lived there, not so much to the modern state and the events that have occurred.

The modern German state of Saxony is not the same as the historical area of Saxony that Bede claims many of the English, Anglo-Saxons, came from. Indeed the modern German state of Lower Saxony is more in the neighborhood of the historical Saxons who took to the seas to go to England. Why the modern state of Saxony is so far removed from the Saxony of the early middle ages is beyond my scope

According to Bede, the Saxons were joined in England with their neighbors, the Jutes and the Angles, tentatively identified with modern Holstein and Jutland. The Saxons also had extensive relations with the Frisians of modern day Belgium and the Netherlands. Indeed, the Anglo-Saxon epic Beowulf even spends a bit of time dealing with not only the politics of Sweden, where the ancestors of the composers are believed to have come from, but also with issues in Frisia.

The Saxons, and Angles, Jutes and other Germanic tribes, who arrived in England brought over a great deal of cultural continuity with them. The artifacts at the Sutton Hoo burial in East Anglia have been identified as sharing commonalities with southern Sweden at the same time, and other similarities in other parts of England to parts of the continent have also been discovered and examined. Other examples of cultural continuity that are firmly established are housing patterns and jewelry, though these changed relatively rapidly. Certain patterns of dress, jewelry designs, and goods found have been drawn to continental, and Scandinavian, contexts as well. However the new comers to the island did not wholly impose their own culture over the existing Romano-British population. Indeed there is quite clear evidence of extensive co-existence, intermarriage, and a gradual synthesis of these peoples that while seemingly dominated by Germanic newcomers, it was in fact far more influenced by native culture than was previously believed.

However what were some of these similarities? The most obvious is religion. The Saxons, and other Germanic tribes that migrated to England, were at this time Pagans, and their religious practices are believed to have belonged to the same tradition that alter gave us the Norse mythologies that were recorded and preserved centuries later. While there is not enough to reconstruct an entire pantheon, cosmology, and specific rites and practices, we do know that certain deities were present in both of these societies, and with the Norse, the exact nature, practices, and role of these deities undoubtedly varied tremendously, and the sources available to us are nothing if not extremely biased against them.

However these cultures diverged somewhat rapidly. The obvious case is in the Christianization of the Anglo-Saxons which took place in the 7th century. The continental Saxons were not Christianized until the following century after they were conquered and repeatedly defeated by Charlemagne and the Franks. Other kingdoms in England also grew more closely related to other insular powers and other realms. The Northumbrians, again according to Bede, were engaged with pretty extensive warfare with the Scots, and farther south in Kent, the ties to the Frankish kingdom were far more pronounced than in other places. Indeed, Bede places the impetus for the Christian mission to England on the Frankish wife of the Anglo-Saxon King of Kent who allowed them to stay and proselytize. These connections soon far outstripped connections to Scandinavia or Saxony in importance for the Anglo-Saxons, and in some cases might have been discouraged during the viking age and the conflicts with Scandinavians.

If you have any follow ups I'd be happy to field them.

---

Sources:

Robin Fleming, Britain After Rome

Guy Halsall, Worlds of Arthur

Peter Heather, Empires and Barbarians

Henry Mayr-Harting, The Coming of Christianity to Anglo-Saxon England

Wow, such a fast And detailed response, upvote for you!