Thank you very much, Nellie and Helen. So it is not as simple as switched captions. However, Roberts in The Queen's Diamonds seems to dispute Nadelhoffer. The Flameng portrait is shown on page 81 with the caption stating "...the Queen is wearing her collier résille necklace, made by Cartier in 1904."
Then on page 105 is this portrait of Queen Mary:
Its caption describes the necklace as "the collier résille made by Cartier in 1904 and altered for Queen Mary in 1926."
Due to the rather loose style of painting used by Flameng I think it is nearly impossible to confidently identify the necklace. However I do think that it could be this one which is being portrayed.
Besides adjusting for the difference in neck size I wonder if the alterations made for Queen Mary included the removal of the laurel leaf borders. I can't tell from the 1913 photos of Queen Alexandra if the leaf border is still intact.
Here is a comparison with a detail of a 1934 portrait of Queen Mary: