Well I can only speak for Germany, but the problem behind that question is universal. Real recycling is only possible if you can determine what material it is made from and it should be one of the basics like Polyethylen or Polypropylen. For that reason, packaging materials that take part in the so called “duales system” in Germany have to wear a sign, defining what material it is made from. These materials can be reused. Packaging material, that doesn’t have this sign can be reused for low-quality uses. We usually produce brown plastic park benches from this “unknown” plastics. A big part of the plastics given to the recycling system is burned, or “thermisch verwertet” (thermically reused, euphemism) in Heatplants, which is not as bad as it sounds, as plastics are made from oil, which gets to a large part burned for energy reasons anyway, so this is just a detour.
So if you throw those picks into the “Wertstofftonne” (recycling bin) it will end up as some low-quality product or it will be burned.
You could try to melt them and cast new ones by yourself But alway non-mixed (except for color, that shouldn’t be a problem), as long as you don’t want UlTorNylex-Picks
On the other Hand, as a dedicated shredder, take your old picks and glue them to a canvas, producing nice pictures. Or glue them to a beat up guitar body and cover it with some kind of resin… thousands of ideas come to mind