For what my opinion is worth, I think that Paisley25 is correct & that it was the Greville emerald tiara referred to.

In the book Prince Harry does not suggest in anyway that his wife wasn't given her tiara of choice. I strongly suspect that, the story the Duchess was refused the use of an emerald tiara, was a media invention.

What Prince Harry was complaining about was that Angela Kelly didn't provide access to the chosen tiara immediately when it was requested so that the hairdresser could practice placing it.