How much of the story is true?

This story is based on more historical fact than you might think.
Thomas More’s first public role was as a pageboy in the household of Archbishop John Morton, Henry VII’s right-hand man, who liked to tell dinner guests that the witty, self-possessed child would one day be a great man. After qualifying as a London barrister, the young More befriended the Dutch humanist Erasmus and a circle of English humanists including Dean John Colet. The group of friends helped Dean Colet to set up a school for city children in the yard of St Paul’s Cathedral — which exists to this day, though in Barnes and Hammersmith – and worked together to set an appropriate curriculum for bright Renaissance children. Thomas More later set up a separate home school along the same lines for his own children, who became famous across Europe for their learning. More had a glittering political career, ending when he resigned as Henry VIII’s Chancellor in 1532. He is also remembered for two books. One is Utopia, a playful and ambiguous description of a perfect land that cannot exist. He explains in the book that this place has been described to him and an assistant he calls his “boy John Clement” during a diplomatic mission by a sailor who likes to tell tall tales. More’s second book is an unreliable but gripping history of Richard III, the last Plantagenet king, who was killed in battle by Henry Tudor when More was five. More’s history formed the basis for Shakespeare’s later play about Richard III, which cast the Plantagenet king as a scheming hunchbacked usurper who murdered his nephews, the Princes in the Tower, so he could steal their throne for himself. More’s book has been denounced in modern times as slanderous victor’s history but remains the basis of most people’s thinking about Richard III. More’s horror of heretics late in his career is well documented, both through his written denunciations of Martin Luther and his Protestant followers and the writings of contemporary friends and enemies.
His adopted daughter Meg Giggs was interested in medicine and was known in the More family for having cured her father of tertian fever after reading the medical writings of Galen. She married the former family tutor, John Clement, a decade after he left the More household to lecture in Greek at Oxford and then train as a physician in Italy. The Clements began their married life at the Mores’ former family home in London, the Old Barge on Bucklersbury Street, from where More (and, briefly, John Clement) was arrested in 1534 before being executed a year later. The Clement family, along with the Mores’ closest friends, the Rastells, later left an increasingly Protestant England for the safety of the Catholic enclave of Louvain in the Low Countries, where they lived out their days. More’s eldest daughter Margaret married William Roper, who hero-worshipped More and wrote an adoring biography of his father-in-law after More’s death. Cecily More married Giles Heron, another child adopted by the family, who was executed not long after More. The only one of the children to escape virtually unscathed from the death of the family patriarch was More’s daughter Elizabeth, whose husband William Dauncey’s political career continued smoothly.
Hans Holbein, a German painter, came to England in 1526 to make his fortune as a portraitist. He spent several months living with the More family at their new home in Chelsea and painted their family portrait. After returning to Reformation Germany in 1528, where the churches were being whitewashed, Holbein was unable to find enough work as a decorative artist to sustain his family. He returned to England in 1532. A second portrait of the More family, which was handed down through generations of Ropers and now hangs at Nostell Priory in Yorkshire, is usually attributed to him even though it is signed “Rowlandas Lockey”. Hans Holbein remained in England till his death a decade after this book ends. In that time, he became the King’s painter and made portraits of many leading courtiers. He died, probably of plague, in 1542 and is buried in one of the churches along Bishopsgate at the eastern end of the City of London. Sweating sickness appeared in England for the first time in 1485, just after Henry Tudor’s victory over Richard III and seizure of the English throne. It was widely believed to be God’s sign that the Tudors were not a legitimate dynasty. It struck half a dozen times while England was under Tudor rule before vanishing forever.
No one knows what became of the Princes in the Tower.