Wow! Thank you, J-F, for these two unknown pictures (unknown to me, at least).
So it makes it possible to confirm that the Kent brooch comes from Queen Mary, and that Queen Mary had this brooch in the early years of her marriage. So the link with the wedding gift from the Milners, even if not still 100% sure, becomes more probable.
By the way, I wonder if your first picture does not give an answer to another mystery. The brooch pinned by the young Queen (and probably then still Princess) Mary on her collar looks very similar to the emerald triple-loop brooch that Queen Elizabeth II has worn a few times, the main and most recent one being the State banquet in honour of the President of the Irish Republic Michael O'Higgins on 8th April 2014. I admit I am not totally sure, though.