Oh! This is such an interesting topic!
For many reasons, I agree with Dave and JohnAF -- that the brooch worn by QEII is probably one of the amethysts ovals from part of the Kent amethyst necklace.
First, I don't think that the diamond surrounds are the same as in the sapphire which Princess Diana eventually used as the centre piece of her pearl choker.
I also think that I can see a hint of gold in the setting surrounding the central stone -- as can be seen in the setting around the oval shaped amethysts in the Kent necklace.
(I am re-posting some of Franck's images)
The surrounds of the amethyst stones in the Kent necklace.
I suspect that the brooch may be one of the "smaller" ovals towards the back of the necklace.
Thirdly, I fancy that I can see a hint of amethyst in a couple of the images Franck posted, but I admit I could be mistaken, as the video we saw is not a high quality version and, of course, colour can be misleading in some older film images.
Another of Franck's images.