According to the press the Countess of Plymouth wore an emerald tiara, necklace and earrings to the Coronation in 1953.
The earliest reports I have found of emeralds in the Plymouth family relate to Irene, Countess of Plymouth. If I have understood correctly she was the mother-n-law of Caroline Rice, Countess of Plymouth.
The earliest mention of the emeralds come from the published lists of the wedding presents given to Lady Irene Charteris in 1921.
She received 2 separate tiaras, one from the groom's grandmother, Walburga, Lady Paget and the other from Sir Reg and Lady Paget.
I haven't found sufficient evidence to state definitely that the tiara worn by Caroline, Countess of Plymouth at the 1953 coronation is the one given to Lady Irene as a wedding gift; however, Irene, Countess of Plymouth was photographed wearing the tiara in the 1930s.