Puns go way back to ancient Egypt, and are found in the bible, and as some of the earliest books translated into English, may well be the source of the "first pun in English".
Beowulf "is one of the very earliest poems in English and its first great literary masterpiece". It was written in Old English sometime between the 8th and 11th centuries.
A footnote in Beowulf: A New Translation for Oral Delivery (2007) says:
Beowulf criticism has devoted a lot of attention to Unferth ... The poet seems to think his name is allegorical and means "bad mind" or "evil mind": this is suggested by the way in which he puns on the names in lines 1165b-66b of the original, where he says that both Hrothgar and Hrothulf had faith in the mind (ferhþ) of Band Mind (Un-ferþ) - obiviously not a very smart thing to do.
The footnote in Beowulf: A New Translation for Oral Delivery (2007) continues:
Punning, incidentally, was very popular among the Anglo-Saxons. Bede's account of Pope Gregory the Great and the English slave boys in Rome is deservedly famous. (Ecclesiastical History, Book II, Chapter 1), and already by the end of the period the incompetent and hapless King Æþelræd ("Noble Counsel") had acquired the nickname Unræd ("Bad Counsel") - whence the name Ethelred the Unready, by which he is know today.
Bede completed the Historia ecclesiastica gentis Anglorum (in English: * Ecclesiastical History of the English People*) by around 731. He wrote in Latin, but an Old English translation was made, sometime in the ninth or 10th centuries.
Geoffrey Chaucer (c. 1343 – 1400), known as the Father of English literature, also used puns in his poetry. "Chaucer’s Cunt" shows some of his puns in Canterbury Tales (1380s–1390s):
Oddly, these statements are followed by quotes from The Canterbury Tales that belie them, for the word that Chaucer uses is not “cunt,” but “queynte.” “Queint,” as a noun, literally means “a clever or curious device or ornament” (Middle English Dictionary) or an “elegant, pleasing thing” (Riverside Chaucer). When used to refer to a woman’s genitalia, it is both a euphemism and a pun.