The Palace has launched an inquiry into how the 17-second black and white film came into the hands of The Sun newspaper.
It shows the Queen, aged six or seven, join the Queen Mother and her uncle Prince Edward in raising an arm in salute as she played alongside her younger sister, Princess Margaret.
It is understood that depending on the outcome of the investigation, the Palace will be looking at issues of copyright and possible criminality.
A source said the royal household is trying to ascertain where the film came from, who it came from and why it was handed over to the newspaper.
A Palace spokesman said: “It is disappointing that film, shot eight decades ago and apparently from Her Majesty’s personal family archive, has been obtained and exploited in this manner.”
The grainy footage from 1933 shows the Queen playing with a dog on the lawn in the gardens of Balmoral, The Sun claims, before she raises an arm to wave to the camera with Margaret.
The Queen Mother then makes a Nazi salute, and after glancing towards her mother the Queen mimics the gesture.
The Queen Mother repeats the salute, joined by Edward, and Margaret raises her left hand before the two children continue dancing and playing on the grass.
Some have criticised the newspaper’s decision to publish the footage, and a Palace source said the pictures should be seen “in their proper context and time”.
The source added: “The Queen is around six years of age at the time and entirely innocent of attaching any meaning to these gestures.
“The Queen and her family’s service and dedication to the welfare of this nation during the war, and the 63 years the Queen has spent building relations between nations and peoples speaks for itself.”
The Sun’s managing editor Stig Abell said the footage was obtained by the newspaper “in a legitimate fashion” and that its publication was “not a criticism of the Queen or the Queen Mum”.
Culture Secretary John Whittingdale said he understands why the Palace is upset by the publication, but he defended the right of the press to make editorial judgements.
Asked if the newspaper was right to publish, he told BBC1’s Andrew Marr Show: “That is a matter for The Sun. It is an editorial judgement. It is up to the press to decide what is and what is not appropriate to print.
“They decided clearly there was a public interest and the British public will judge whether or not they were right.
“Sometimes editors have difficult decisions. Sometimes people will think they are right, sometimes wrong.
“We have an independent press regulator which is available.
“But I can understand, in this particular instance, why the Palace were upset by it.”