You could say that the character of King Lear relates to Elizabeth and James I because both had a reputation for preferring fawning courtiers over true advice. With Elizabeth the courtiers were usually men seeking her favors. With James I, many say the same is true, he preferred good looking young men who fawned over him, protesting their love. But I really think that Lear relates more to Henry VIII, the father of Queen Elizabeth.
Henry also had three children, by three different women and he played them against each other to the detriment of the kingdom and was more interested in lip service than in true loyalty. He even had his best friend, Thomas More, put to death for a treason which was only verbal. Henry could not abide anyone disagreeing with his smallest whim. He divorced two of his six wives and beheaded two for disloyalty. By the time Henry died he had managed to alienate anyone who had ever truly cared about him so that his death was essentially unmourned. While he did not abdicate his throne, the succession games he played with his three children wreaked havoc on the country. Edward's reign was dominated rival regents. Mary Tudor (not the Mary, Queen of Scots referred to above), a bitter spinster when she came throne spent her reign killing Protestants and suffering hysterical pregnancies in her desperate and unpopular marriage to the King of Spain. Thus Elizabeth is more like Cordelia as she is the true and responsible heir.