Quite a bit, actually. Graeco-Roman mythology was a significant part of the education curriculum. Much of the educated elite would have been broadly familiar with ancient Greek mythologies through its Latin form, albeit overlaid with a Christian point of view.
In High Medieval England, an anthology of six works known collectively as the Liber Catonianus (aka Sex Auctores) became a standard textbook of sorts. Widely used in English schools, they notably included:
Together, these texts provided medieval English students with a somewhat surprisingly rich education in ancient Graeco-Roman religion and culture.
Both Claudian and Statius were Ancient Roman poets, while the Eclogue is dated to between the 9th and 10th centuries, written perhaps by a Saxon theologian called Gottschalk of Orbais. A translation is available here, which gives you an idea of what a 13th century student in England would've known. It contains many Graeco-Roman myths familiar even to a modern audience, for example the myth of Proserpina. Moreover, its status as a standard text for instruction means we can expect the educated class to be reasonably familiar with its contents.
Much of the myths related by the Eclogue would also have been widely available in the form of two more sources: Metamorphoses by Ovid, and the works of Virgil. Both Roman poets drew considerably on Graeco-Roman mythological traditions, and were held in extremely high literary esteem by Medieval Europeans. Virgil's Fourth Eclogue was even perceived as a messianic prediction of the birth birth of Jesus. Their popularity during the Middle Ages was a significant influence on writers of the time: Metamorphoses clearly had a big impact on the works of Chaucer, for instance. Given their status as Medieval pop culture blockbusters, most educated people in the 1200s were likely familiar with their works.
Note that all of this were Latin, at the time the lingua franca of Western Europe including England. The relative abundance of information on Classical mythology were likewise transmitted from mainly Romans via Latin poetry. Whatever knowledge of Greek traditions existed was, therefore, filtered through a Roman viewpoint.
Orme, Nicholas. Medieval Schools: From Roman Britain to Renaissance England. Yale University Press, 2006.
Hunt, Tony. Teaching and Learning Latin in Thirteenth-Century England. Boydell & Brewer, 1991.
Wallace, David, ed. The Cambridge History of Medieval English Literature. Cambridge University Press, 2002.
Apart from Latin works such as those mentioned above, classical texts were largely "lost" to Western Europe during the Middle Ages, due to the knowledge of Greek becoming very rare in the Latin West. In the 13th century, however, a rediscovery was just beginning to occur. Many classical texts had been translated into Arabic following the rise of Islam. Starting with the reconquest of Toledo in 1085, a large collection of such "lost" works fell into European hands. The Arabic translations were relatively easily re-translated into Latin, and in this way classical knowledge spread throughout Latin Europe again.
Of course, these were still transmitted in Latin. So if any text on Greek mythology reached 13th century England in this way, literacy in Latin would still be required to understand it first hand.