Tolkien was a professor of Anglo-Saxon at at Pembroke College from 1925 to 1945. He was strongly influenced by Old English literature such as Beowulf and favoured especially the Mercian dialect. Tolkien wrote in Old English several texts of his legendarium, which he (fictionally) attributed to Ælfwine such as the Earliest Annals of Valinor in The Shaping of Middle-earth; although they are commented and edited by Christopher Tolkien, the translation is not given.
For Lord of the Rings, Tolkien used several Anglo-Saxon names and words, which represented Rohirric, the language of the Rohirrim. This simulated the archaic sense the Hobbits (who spoke Hobbitish Westron) felt with the Rohirrim.
However the relation of Old English and English is not the same as with Rohirric and Westron, since the latter descends from Adûnaic, not Rohirric.