The film, obtained by The Sun newspaper, shows the six or seven-year-old princess joining her mother and her uncle Prince Edward in raising an arm in salute, as she played alongside her younger sister, Princess Margaret.
A Palace spokesman says it is "disappointing" the newspaper has exploited a film shot eight decades ago, and from Her Majesty's personal family archive.
"It is disappointing that film shot eight decades ago and apparently from HM's (her majesty's) personal family archive has been obtained and exploited in this manner," a spokesman for Buckingham Palace said in a statement.
The group, which also included the queen's sister Princess Margaret, were apparently being encouraged by the queen's uncle, the future king Edward VIII.
The precise nature of Edward's links to the Nazis are still debated in Britain, with some historians accusing him of being sympathetic to Adolf Hitler's regime.
The Sun defended its decision to release the images, saying they offered "a fascinating insight in the warped prejudices of Edward VIII".
