Re. Queen Alexandra's turquoise tiara.
According to several news reports in 1902, King Edward gave a turquoise and diamond tiara to Queen Alexandra as a 58th birthday gift. Two examples:
“Monday was Queen Alexandra's 58th birthday and amongst the presents she received was a turquoise and diamond tiara from King Edward, and a jewelled tortoise -shell box from the King of Portugal". 07 December 1902, Lloyd's Weekly Newspaper .
“...and was the recipient of an immense number of beautiful presents, including a magnificent turquoise and diamond tiara from King Edward, and jewelled tortoiseshell box from the King of Portugal”.
06 December 1902, Bucks Herald, Aylesbury .
It could be the turquoise and diamond tiara now worn by QA's great granddaughter Princess Astrid of Norway.

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I published an article today about Empress Frederick’s Pearl Corsage Brooch, HERE, to mark the 180th Anniversary of the Birth of Victoria, Princess Royal and Empress Frederik.
This is a jewel on which I have been gathering research for years and while I was initially sceptical (as suggested by a few sources including the British Royal Jewels Account on Instagram ) that it was the same brooch that now belongs to Princess Alexandra, after meticulous observation, I do think that it is the same piece.
The biggest question is how it went from Queen Sophia of Greece to Princess Alexandra? I believe that Queen Mary or Princess Marina, Duchess of Kent acquired the Brooch from the Greek Royal Family anytime between the 1920s and the mid-1950s, possibly during one of their periods in exile. I would think it's Queen Mary, as Princess Marina did not have the funds to acquire jewels after the late 1930s, but she did have the close connections to the main Greek Royal Family. I did read recently that Queen Alexandra's Turquoise Tiara worn in the Norwegian Royal Family was originally the Turquoise Tiara of Queen Sophia of Greece, and was acquired by Queen Alexandra sometime in the 1920s before passing to Queen Maud of Norway. Could it be that Empress Frederick’s Pearl Corsage Brooch was also acquired by the British Royal Family at the same time?