I think it is unlikely we will see Royals buying jewels at auction as the purchase is instantly traceable and dependant on the price paid can cause criticism about wasteful/excessive spending.

I know the Crown Princess of Denmark bought a tiara at auction a few years back but I believe it cost just a couple of thousand euros and so didn't cause much of a ripple.

I think any future royal purchases will be via private sale, which most of the major auction houses facilitate.

I am sure if the Duchess of Sussex wanted her own tiara she could enquire of Sothebys or Christies and they would source one for her privately.