There is much speculation about the identity of King Cole, but it is unlikely that he can be identified reliably given the centuries between the attestation of the rhyme and the putative identities; none of the extant theories is well supported.
William King mentions two possibilities: the “Prince that Built Colchester” and a 12th-century cloth merchant from Reading named Cole-brook. Sir Walter Scott thought that “Auld King Coul” was Cumhall, the father of the giant Fyn M’Coule (Finn McCool). Other modern sources suggest (without much justification) that he was Richard Cole (1568–1614) of Bucks in the parish of Woolfardisworthy on the north coast of Devon, whose monument and effigy survive in All Hallows Church, Woolfardisworthy.
Source: Old King Cole – Wikipedia