Shakespeare's commonly-recognized sources include no Old English works (though they do include Chaucer). His sources for English history, like Holinshed's Chronicles, were roughly contemporaneous (and are recognizably the same language we speak today, once you get used to the non-standard spelling.)

The noted sources for the Chronicles (see bottom) are largely in Latin, French, Greek, or roughly contemporary English, but a handful of Anglo-Saxon texts are mentioned, so it seems that someone on the Hollinshed team, or an antiquarian they had access to, could read it.