The following information was provided by Sam Hearn and Ian Franklin, who work at Hampton Court Palace and are "keen to ensure that every story can be traced back to its source and is not simply the invention of some imaginative writer."
The palace was built (or at least begun by Cardinal Wolsey). Building work began in 1514 but the first phase of construction continued until 1522. Work restarted in 1525. When Henry VIII swapped his Richmond Palace for Wolsey's Hampton Court in 1525.
The two men haunting Fountain Court are now thought have been workmen who died when there was a major collapse during of Sir Christopher Wren's rebuilding of the Palace in the late seventeenth century.