Buckingham Palace staff have been relatively and characteristically restrained in their response, saying that they are disappointed that the Sun has seen fit to intrude into the Royal family's private life. But others have not been so reticent in their comments. The BBC, for example, has speculated that the Palace may try to sue the Sun for infringement of copyright by releasing the film clip via their website. Stills taken from the film which appeared in the paper would not constitute fair dealing for the purposes of news reporting because photographs are excluded from this exception.
Apart from the political and public relations issues with the Queen starting a private civil case of this nature, there are also considerable legal problems with this course of action, should it be seriously considered.
However this is not the case for photographs created prior to 1 June 1957. Section 21 of the 1911 Act afforded special treatment to photographs, namely that the term of protection was fixed at 50 years from the making of the original negative, and then ceased. This applied equally to published and unpublished photographs. What is more, although the film (reputedly shoot in 1933 or thereabouts) would still have been protected under section 21 when the 1956 Act came into force, the transitional provisions (in Schedule 7) of the later Act did not change the provisions for photographs made before 1 June 1957, and so copyright in the film would have expired around 1983, some time before the current 1988 Act appeared on the statute books.
Hang on though, what about Crown Copyright? The 1911 Act, although not using the exact term, introduced what we think of as Crown Copyright, which lasted for fifty years from the date of first publication of the individual work. Since this film (we presume) has never previously been published, that would mean that the copyright term has not yet commenced (unauthorised publications such as that by the Sun newspaper don't count). But could the film attract Crown Copyright? The wording of Section 18 of the 1911 Act says:
"Without prejudice to any rights or privileges of the Crown, where any work has, whether before or after the commencement of this Act, been prepared or published by or under the direction or control of His Majesty or any Government department, the copyright in the work shall, subject to any agreement with the author, belong to His Majesty, and in such case shall continue for a period of fifty years from the date of the first publication of the work".
It seems unlikely that the then King, George V, would have directed that this film be made, although it is possible. From the circumstances and individuals shown in the film, it seems probable that the person filming the goings-on was the King's second son, Prince George, who later became King George VI. Given the private nature of the film, and following the dicta of the court in the case of the Black Spider memos, this private activity would not constitute the duties or authority of the 'Crown'. Arguably had a decision on this sort of issue been required back in the middle of the last century when deference was rather more in vogue, things might have been handled differently (see the case of Prince Albert v Strange 1849 discussed by Jeremy here).
However I suspect that all this will be hypothetical, and that no such copyright case will ever be contemplated, let alone go forward to litigation. But if it did, the Sun would undoubtedly advance the little-developed defence of public interest under Section 171(3). The Sun is of course no stranger to the public interest defence in a copyright case, since it was argued in the case of Hyde Park Residence Ltd v Yelland and others  which the Sun lost on appeal.