Not too many polymers in a modern silver flute, except the pads and the oil. The irish flute,
however is a transverse flute which is still made of wood.
Nowadays flutes are generally made of silver or silverplated nickel. But originally they were made of wood. A wooden flute has a distinctly different sound from a metal one. The natural plant fibers in the cellulose give it a "reedy" sound somewhat like a recorder. Some wood flutes have keys and some of them don't. And modern flutes have very complex key mechanisms with felt pads which cover the holes. Some musicians still perform on flutes made of wood because they simply prefer that "wood" sound or because this sound is more appropriate for older music.
The ABS resin piccolo has a metal mouthpiece,
while the wood one can have a wood or metal mouthpiece.
A piccolo is a lot like a flute, but even higher! This little instrument is made from wood or ABS resin and can have a wood or metal headjoint. And there are also piccolos made completely of metal - silver or silverplated nickel like flutes.
The joints in wood flutes and piccolos are - as with other woodwind instruments - made from cork, which must be oiled so it will remain soft. As with flutes, the modern piccolo has a complicated arrangement of metal keys. But this was not always so. The oldest piccolos were not transverse flutes at all. They were French instruments called flageolets or flauto piccolo in Italian, and much more like sapranino recorders, blown like a whistle, some with keys and some without. And they were made, of course, from wood.
A fife, on the other hand, is much like a modern wooden piccolo, except with only six holes and without keys (Fifes also come in metal). You'll see this instrument in lots of Revolutionary War re-enactments. For some reason back then soldiers liked high-pitched woodwind instruments.