Anne-Marie married the then King of the Hellenes. This took her out of the line of succession and that includes her issues.
This is correct. By marrying a foreign dynast, former queen Anne-Marie of the Hellenes, has lost not only her right of succession to the Danist Throne but, also, her rank and title. Thus, she is no more princess til Danmark , but because she was born a princess of Denmark prior to the introduction of the Act (1953) she may, by courtesy, be referred to as princess af Danmark.
Similarly, former King Constantine of the Hellenes, was born (1940) prince of Greece and til Danmark, as a direct patrilinear descendant of King Christian IX. But, since the introduction of the Act of 1953, he has lost his rank and title and may only by courtesy be referred to as prince af Danmark.
Because they were all born after 1953, the children of Constantine and Anne-Marie are not princes or princesses of Denmark in any way, type or form, not even by courtesy. In fact, it would be disrespectful of the Constitution of Denmark for anyone to address them as such.
Can the Queen, or the next ruler, declare the Berleburg Children HH, Prince/Princess of Denmark?
The children of HRH Princess Benedikte are not princes either til or af Danmark, because when she married in 1968, her late father, King Frederick IX decreed that in order for them to become princes of Denmark and acquire succession rights, they would need to be raised in Denmark. This requirement was not met.
Of note, the 1953 Succession Act became part of the Constitution of Denmark. The Act specifies among other things that "when approving a marriage, the monarch can impose conditions that must be met in order for any resulting offspring to have succession rights."
Therefore, it seems that a decree by the current monarch would not be able to reverse the situation with HRH Princess Benedikte's children. Currently, only an amendment to the Act could achieve that which appears to be an elaborate process and not justified since there are plenty of people already available to inherit this old and prestigious Throne.
The "official window of Denmark":
Denmark - Official website of Denmark
says
I find it interesting that it makes reference to a "narrow sense".
Who are all the members of the Royal House are in a "wider sense" ?
And who ranks above whom ?
Who is "Princess Elizabeth", and how is she related to the Queen ?
Actually, there is no such thing as narrow, wide, or wider sense, as far as the Danish Constitution is concerned vis-a-vis the royal family. Legally and constitutionally, the Royal House (synonymous with the Royal Family, RF) consists only of the sovereign and her/his spouse, any living consort of a deceased monarch (as was the case with HM Queen Ingrid after King Frederick's death), and those members who have rights of succession and as such hold rank and title of prince/princess, along with their spouses. Thus HRH Princess Benedikte and her husband are members of the RF.
Importantly, according to the 1953 Succession Act, all those relatives who have no rights of succession, are not members of the Royal Family. Thus, HRH Princess Benedikte's children who have no Danish rank and title, are not members of the RF. Likewise, the former queen of Greece, her husband and her children have no Danish rank or title and are not members of the RF.
However, queen Anne-Marie, king Constantine, their children and the children of Princess Benedikte are still very close relatives of The Queen and members of her family, the term family considered here only in a private/personal sense. Thus, when HM The Queen uses the expression "in a narrow sense", she is just being polite, not meaning to alienate her sister queen Anne-Marie, former queen of the Hellenes and her children as well as the children of Princess Benedikte.