I doubt it will be non-metal. The combination of density and ductility for inertia and 'engravability' aren't really available in anything else that I can think of.

There has been talk if using some kind of nuclear waste (depleted uranium?) for tank shells.

How feasible would that be for hunting and target shooting? It is said to be very heavy, which means a small projectile at a high speed. Does this contain a health risk?

Not so much a new "bullet" material, but with the SIG Military contract, not only did they manufacture a few new Firearms but they created a new hybrid cartridge (Caliber: 277 SIG FURY / 6.8 x 51)

https://www.ssusa.org/content/sig-sauer-announces-commercial-version-of-u-s-army-ngsw-mcx-spear-277-sig-fury-ammo/

There has been talk if using some kind of nuclear waste (depleted uranium?) for tank shells.

How feasible would that be for hunting and target shooting? It is said to be very heavy, which means a small projectile at a high speed. Does this contain a health risk?

It's used in kinetic munitions because of it's density but also because of its hardness. Suspect that it would be too hard to properly engrave in a rifle so it might need to be combined with something softer that could provide a bearing surface. Beyond that you couldn't convince me to eat animals whacked by a bullet made of the stuff unless I was a 6 or 7 days hungry at least. 

There was on some site some talk of titanium and a epoxy that made for a very, very high BC - SD bullet. Some experiments were done, but the cost of titanium was prohibitive and the super SD was wasted as it was  only really suited for target shooting. No hunting application as it could only be made as a stay solid or shatter bullet.
There was some later mention of lead titanium attempts to up the SD of hunting bullets.
Have not thought of this subject in like 10 years, I stumbled onto matter when trying to get most out of a .223 for Cody to start hunting with. I ended up using Sierra 63 gr bullets.

I hunted with a guy that used tungsten bullets.

They cost the earth, but tungsten is 1.74 times the density of lead, so you can imagine the momentum and penetration of such a bullet.

They looked like monos that were quite hollow inside with a big plastic tip to expand the metal that I presume is much harder than lead.

No idea what they do to your barrel?

I hunted with a guy that used tungsten bullets.

They cost the earth, but tungsten is 1.74 times the density of lead, so you can imagine the momentum and penetration of such a bullet.

They looked like monos that were quite hollow inside with a big plastic tip to expand the metal that I presume is much harder than lead.

No idea what they do to your barrel?

No sabot to protect the barrel?

Tungsten is 9 10ths as hard as diamond. I have to assume that the bullets in question had some or other jacket of softer metal that would engrave and not nuke the barrel.

Maybe a tungsten alloy? Pure tungsten shatters on impact - an engineer once showed it to me by dropping a piece on the floor.

Depleted uranium (DU) use explained

https://youtu.be/9W_nMRbIlZI

https://orau.org/health-physics-museum/collection/consumer/depleted-uranium/penetrators.html

There was on some site some talk of titanium and a epoxy that made for a very, very high BC - SD bullet. Some experiments were done, but the cost of titanium was prohibitive and the super SD was wasted as it was  only really suited for target shooting. No hunting application as it could only be made as a stay solid or shatter bullet.
There was some later mention of lead titanium attempts to up the SD of hunting bullets.
Have not thought of this subject in like 10 years, I stumbled onto matter when trying to get most out of a .223 for Cody to start hunting with. I ended up using Sierra 63 gr bullets.

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;D  ;D  😮  😮 me and facts - perhaps it was tungsten, not titanium  :-  - some supper heavy stuff, thought it was same as my watch ??

I found Powel River Laboratories and DRT make a compressed tungsten fragment core bullet in copper cup, but info on these is scant, other than the fact that they were very heavy for their size, and killed fast.

A question on toxicity of tungsten gave the following interesting facts.

https://wwwn.cdc.gov/TSP/ToxFAQs/ToxFAQsDetails.aspx?faqid=805&toxid=157

A bit embarrassed about my Titanium post, not one of my dear mates corrected me  😮
Ya well anyway, when I was trying to up the effectiveness of the .223 as a hunting bullet, I came across a few posts of guys using tungsten and complaining about the price and apparent unobtainable status in recent years. There was talk of resin,epoxy and press mold forming bullets.
I have not tried to find the article again, but do recall that tungsten was no longer available mail order in the states  and when found it was stupid costly. The bullets had issues staying together and much heavier bullets were shot through standard twist barrels.