However, I have not been able to find any evidence of them wearing or receiving the order.
As mentioned below, only seven badges were produced in 1937.
Of the four mentioned by Vickers, only two attended the 1937 Coronation, Queen Maude of Norway - the King's Aunt - and Princess Alice, Countess of Athlone - the King's first cousin once removed and Aunt. Both wore the Royal Family Order of King George V, but not King George VI.
Queen Maude of Norway.
Princess Alice, Countess of Athlone.
The other royals at the 1937 Coronation with George V's Family Order that did not receive that of George VI were Lady Patricia Ramsay, Princess Arthur of Connaught, Duchess of Fife, and Maud Carnegie, Countess of Southesk.