I have had a look at the QD and it seems no screwdriver is needed to swap the jewels in the Vladimir. But note the frame was completely remade in 1988.
In one pic it is quite clear that an emerald is suspended on a long hook very like the long hooks we see for earrings through pierced ears. The hooks go through tiny eyelets at the bottom edge of the ribbon laced through the tops of the hoops.
You can get the idea from the fourth last emerald to the right here.