The oldest pieces in the Royal collection could be Queen Anne and Queen Caroline's pearl necklaces, dating back to the first part of the 18th century:
The oldest brooch in the Royal collection could be the set of three brooches, part of the Duchess of Kent's Amethyst parure, which probably dates back to the late 1810s (Prince Edward, Duke of Kent, and father of Queen Victoria, died in 1820, so I guess his wife received this parure before she was a widow).
The round Cambridge emerald brooch might be a serious contestant, though. We know that the emeralds were acquired by Augusta, Duchess of Cambridge at a lottery in 1818. Unlike the other jewels made with the Cambridge emeralds, this brooch was not made by Queen Mary for the Delhi Durbar in 1911, but already existed at the time of Queen Mary's mother, the Duchess of Teck. So maybe the brooch was made early in the 19th century?