Here is what I think about the loss of German jewels:
After WWI, in 1919, the monarchy was abolished in Germany. As a result the royal and noble family lost their power and official status, but they were allowed to keep their private wealth. All the lands, castles, artworks and jewels were kept and the lifestyle of those families did not change so very much. Keep in mind that WWI was basically fought for foreign territory, so that the castles were still intact.
Sesa, I never heard that much aristocratic jewelry ended up with the Nazis. The mainly Nazis plundered Jewish possession in abominable ways. But aristocratic possessions were kept intact unless the owners were caught for being involved in the resistance.
It was at the end of/after WWII that the Russian occupants systematically looted aristocratic possessions. Lands, castles and artwork in the territories which had become Polish or were now the GDR were confiscated in big style. The previous owners often had fled to the west and were able to bring only few, if any, possessions with them. And what could be saved often was sold later to use the money for building a new life in the west.
Those families whose former territories were in Western Germany were better off, their possessions were not confiscated. But during the bombings some castles were destroyed or damaged, artworks and furniture were sometimes lost as well. Money was needed for survival and later also to rebuild the castles. I imagine that some jewels were sold for that purpose.
The jewel chest of the Hesse family greatly suffered from a theft. Some US army members had discovered hidden boxes of jewels and secretly transported them out of the country. But this was the act of a couple of criminal individuals. They were acting only for themselves and not on orders from their superiors. When they were discovered they were properly prosecuted by the authorities. But only part of the loot was found and could be given back to the proper owners. So this theft was an exception and nothing that happened to many families.
In later years the reasons for selling jewels were the same as in other countries: tax bills, failed investments, debts, expensive repairs of the castles, the wish to finance good education for the children etc. might have lead to jewel sales.
. Imo the shape and size of the stones look identical, they could be three of the oval and two of the square emeralds from the big tiara. It wouldn't surprise me if some elements were detachable. But with black and white pictures we can't be sure that they are even emeralds. So it is only an unproven theory.
