In 1980, Queen Elizabeth met with Pope John Paul II at the Vatican as part of a State Visit to Italy. She has also met Pope John Paul II, Pope John XIII, and Pope Pius XII at the Vatican (the latter when she was a princess) in private audiences. Pope John Paul II and Pope Benedict XVI met her during their visits to the British Isles, in 1982 and 2010, respectively.
This is not a State Visit--the Queen and the Duke are having a private luncheon with the President of Italy and then going to the Vatican to meet Pope Francis. They will meet at the Santa Marta guesthouse and not in the Apostolic Palace.
There are many pictures on the web of Queen Elizabeth's earlier visits, wearing a black dress, black closed shoes, and a black veil--but the veil or head covering is not technically required.
Because of the English Reformation, the history of English and Vatican diplomatic relations since the 16th century has been uneven, to say the least. Before Henry VIII"s break from Rome (Pope Clement VII), English monarchs sent ambassadors to the Papal States, as the popes of that era were secular rulers in addition to being the Bishop of Rome and the Vicar of Christ. Henry VII and Henry VIII additionally had "Cardinal Protectors" who addressed ecclesiastical matters like the appointment of bishops. England had a long history of links to Rome and those links had been strong for centuries--Saxon and Mercian kings began the tradition of giving Peter's Pence, which of course ended during Henry VIII's reign.
Mary I and James II both re-established some diplomatic ties to the Vatican, but her death and the Glorious Revolution, respectively, ended those brief periods. Informal diplomacy between the Great Britain and the Pope took place during the French Revolution under the argument that the enemy of my enemy is my friend--in opposition to the French Revolution and the First Empire under Napoleon.
The first post-Reformation ambassador was assigned to the Vatican in 1914 at the beginning of World War I--and he was a Catholic, Henry Howard. His official title was "His Majesty's Envoy Extraordinary and Minister Plenipotentiary on a Special Mission to His Holiness the Pope". Things were stable for awhile and then in the 1930's troubles between Church and State in Malta led to another long break in higher level diplomacy at least. The Troubles in Northern Ireland also strained relations and an ambassador-level official was not appointed again until 1982, the same year that Pope John Paul II visited Scotland, England, Ireland, and Wales.
The current British ambassador to the Holy See (not the Vatican City-State) is Nigel Baker. The current ambassador from the Holy See to the Court of St. James--that is, the current Apostolic Nuncio--is Archbishop Antonio Mennini.