People of this period knew and copied the classics. Most of the stories for which we have the full texts are "high art". Medieval and Renaissance authors, poets, and playwrights quote the same sources repeatedly.
The most influential native British work is the Arthur cycle of stories. It has roots in Welsh poetry, with French and German influences, but became solidly English with Malory's Le Morte d'Arthur.
Though the strongest early poetic traditions are in Wales and Ireland, England developed a literary school. Works from the medieval period that influenced Renaissance readers include the collections Gesta Romanorum and The Canterbury Tales.
Before printing, an original work survived only if it quickly became popular enough to be copied widely. Other works are famous to us today, but we know only one or two copies. Even a masterpiece was not influential if it was not circulated. The most famous example of this category is Beowulf. Some of these works survived as oral tradition, which meant they began to change immediately. Beowulf became the story of Saint George and the Dragon.
Wales and Ireland each had rich myth cycles. The most important Welsh myths were collected as the Mabignonion. In addition, there is a large body of surviving poems written down by hundreds of Welsh poets. Some of these are signed with the names of the heroic figures Taliwsin and Merlin. The Irish myths were assembled into many different books and collections. I give one modern collection of the classic myth cycles in my bibliography.
I have not found examples of a strong Scottish written tradition before the Renaissance. Scotland has a rich oral tradition, and a later literary tradition.
There was also an English tradition of historical annals, which Shakespeare and others drew heavily on. in addition, Renaissance English writers knew older non-British European writings. Every educated person knew the Roman poets, playwrights, and historians. Many knew some of the Greek material as well. The French romances of Roland and the Roman du Rose were influential. The event that defined the start of the Renaissance was the fall of Constantinople, which led to Arabic material becoming known in Europe. al Tannukhi was one Arabic collection of stories.
A large oral tradition of folk tales and fairy tales thrived. Unfortunately, being oral, we don't have written records of it until a later period. We do have hints in written works that tell us oral storytelling was alive and well. Writers who reference a folk tale, without telling the tale, expected their readers to know the story and get the joke.
In 1590 Thomas Nash quotes "Fy, fa, fum, I smell the bloud of an English-man." Some scholars say Jack and the Beanstalk is the only fairy tale we are sure originated in England.
Chaucer's Wife of Bath's Tale is a variant on the Arthurian story we know as Dame Raglan or The Loathly Lady. The Knight's Tale also contains Arthurian references and flavors.
Robert Rypon, a contemporary of Chaucer, mentions Cinderella. Peele's play The Old Wives' Tale in 1594 contains The Three Heads in the Well. Reginald Scot in the 1540s mentions A Maid Who Fetched Water in a Sieve. Ben Johnson in the 1590s lifts elements of The Miser and His Wife from Lucian's 9th Dialogue and inserts them into two of his plays, Volpone and The Case Is Altered. In 1621, Tom Thumb is earliest fairy tale in print in Britain. It appears as an illustrated broadside.
Shakespeare shows us every part of life, and he expects people in every station of life to know their fairy tales. He uses at least 6 story references:
Mr. Fox appears again in The Faerie Queene. Book iii. Canto xi. St. 54:
And as she lookt about, she did behold,
How over that same dore was likewise writ,
Be bold, be bold, and every where Be bold,
That much she muz'd, yet could not construe it
By any ridling skill, or commune wit.
At last she spyde at that roomes upper end,
Another yron dore, on which was writ,
Be not too bold; whereto though she did bend
Her earnest mind, yet wist not what it might intend.
The paired mottoes "Be bold, be bold, but not too bold" appears in a similar context in Mr. Fox. However, The Faerie Queene was published in 1590, before the first printed version of Mr. Fox. It is possible this is an example of a fragment from a literary work drifting into a folk tale.
The folk stories and fairy tales repeat themes of fright and comeuppance. "Pride punished" occurs in at least half the stories I have found in this period. Most of the rest are scary stories. When Elizabethan writers describe storytelling, again and again they use as an example Raw Head and Bloody Bones.
Later and secondary sources
Folklorists began to hunt, preserve, and publish oral tales in the late 1700s. Here are some of the valuable collections: