The most famous (if not necessarily the most popular) instrument used by these peoples, especially Saxons, would be the Saxon lyre, known in Old English as haerpa.

Archeological finds of such lyres date from 5th to Xth century, but almost certainly similar instruments were used prior, and after that period. The most famous one was found in Sutton Hoo and dates to 7th century, which is exactly the time, place and culture you are asking.

As far as can be guessed, Saxon lyre was a 5-7 string instrument that was strummed, while some strings were silenced (a sort of inverse technique of the one used with modern guitars). In the video below you can see an amateur presentation of how it might have been used: https://www.youtube.com/watch?v=2VO-lbCflKY

SOURCE: Mitford,Rupert,Myrtle.: The Sutton Hoo Lyre, Beowulf, and the Origins of the Frame Harp.

If you are interested in knowing more about such a lyre, there is a good artcile published on Vassar College website: https://www.cs.vassar.edu/~priestdo/lyre.html

As for other instruments, there is a large number of finds of flutes, pipes and even panpipes from Early Medieval period from all over Europe. Im not however an expert enough to say which of those were Saxon, and which were brought later by the Vikings.