Thank you very much, Beth, for providing these very interesting links.
I second Franck's comments: it is very interesting to note that this article mentions the Greville emerald tiara as being part of the Greville bequest. If this article is not wrong, I wonder why this very beautiful tiara (a picture of it is published in Vincent Meylan's book about Boucheron), with a very large and impressive central cabochon emerald, has never been seen in public since it entered the royal jewel collection. I am not sure it would be a good match to the Greville emerald necklace (whose emeralds are facetted, not cabochon), but I think it would be a very good complement to the Delhi Durbar emerald necklace.
If King George VI had reservations about his wife wearing the Greville jewels, it was probably in reference to the WW2 context. In those dark, terrible years, when so many people of the UK and of the Commonwealth were facing death, either on the battlefield or through bombings onto their homes, he (rightly) felt that any display of major pageantry would have been unappropriate in such circumstances.
And even the years immediately following the victory in 1945 were still tough times, with strict rationing of food, energy, fabric... Probably the King's sense of "public relations" (as we say nowadays) warned him against possible negative reactions against the simultaneous appearing of too many lavish jewels. The Royal tour to South Africa in 1947, with Princess Elizabeth's 21st birthday, gave a good (i.e. acceptable) opportunity to display again the usual pageantry of the British royalty.
I wonder if another reservation from King George VI could have come from the fact that such a large and impressive jewel collection was given by a private person, and thus could have made the Royal family beholden to this particular person, thus threatening the King's independance and authority. But I do not think this argument is very convincing. It was quite customary for the British Royal family to accept jewellery gifts from the British nobility (see the numerous wedding gifts presented to Queen Mary by all the grand aristocratic families!). And in the case of Mrs Greville, it was known that she was a personal friend of the King and Queen, and that she had no immediate heirs, so I guess there was nothing wrong in accepting such a bequest.
Or could it be a mere (and more intimate) feeling of jealousy from King George VI? From the early months of their marriage until his death in 1952, he used to present regularly nice jewels to his beloved wife... but he never gifted her with such impressive jewels like the Greville Honeycomb tiara, the Greville five-row festoon necklace or the huge Greville diamond drop earrings (I think his "biggest" gifts were the Cartier Halo Tiara, the Aquamarine Pine Tiara or Queen Elizabeth's Coronation riviere). Maybe he could have felt that his own personal gifts were "outranked" by such a large and lavish bequest???
Of course, this is my very own speculation. Unless we discover documents in the Royal Archives about the Greville bequest, we will never know what were King George VI's feelings regarding this bequest.