Depleted uranium has been used as an industrial material at least since the mid 1940s. Like any other toxic material, it requires proper safety procedures for handling and processing, but it's a pretty "routine" thing in those shops that have a need to handle it.
The main source of DU has been from the "leftovers" from production of weapons grade uranium. The standard cutoff point for discarding uranium at the extraction processing plants is at 0.75% fissile isotope content, which is significantly lower in radioactive emission than many naturally occuring ores, and in fact is well below the level of ionizing radiation emitted by one rock that's a *"state marker stone" along the stairs to the top of the Washington monument.
* They all look a bit like tombstones, but that's a bit beside the point.
One common use for DU has been in counterweights in aircraft control systems. I haven't seen recent advisories, but some years ago the stuff could be used only in closed compartments not occupied by personnel in normal operations, or with a minor bit of shielding almost anywhere. An application I was asked to review (decades ago) would have required a stainless steel wrapper about 0.08 inch thick, because it was in an open area about 15 inches from the pilots personals, and by the time you wrap high-density DU in low density CRES, the total effective density isn't any better than lead so I got to shoot down the plan. If they'd actually needed a big weight it probably would have gone into that single-engine prop plane, and been sold to thousands (well dozens at least) of wealthy playboy doctors and lawyers.
Casting, rolling, and machining requires safety provisions not much different than a number of other toxic materials, and while there are safe-distance and shielding requirements, it's the chemical toxicity that is the principal concern - not radiation.
DU armor, as I've seen it described, is encased, with each individual part separately and completely sealed. Rumor says that the popular encasing material now is teflon, probably along with some other materials. If anything in an armored vehicle burns, the teflon is about as much a chemical hazard as the DU.
The main use of DU in weapons has been essentially as a very high speed flying cold chisel intended to punch a hole when it hits something. The properites of DU are not a lot different than can be obtained with tungsten carbide, which has also been used, with a couple of exceptions. The first difference is that DU is really cheap and Tungsten Carbide is really expensive. ($1/lb vs $30/lb in 1974, and the same ratio probably is still good.) The second major reason is that DU can be easily cast, rolled, and/or machined, and Tungsten Carbide can't. Almost a side benefit is that DU is pyrophoric.
Pyrophoric means that DU will "strike sparks" when it hits something, much like a lighter flint. Finely divided bits will "burn" and it's not uncommon to see fine metal turnings smoldering during machining. The bulk material will not sustain "burning" unless external heat is continuously supplied from something else that gets lit from the sparks. During penetration of steel armor, about 40% of a DU cold chisel is vaporized, and about 10% of that vapor is small enough to remain suspended for a significant time, and to be breathed by anyone who happens along. The remaining "vapor" that's generated by the impact will settle out of the air within a few minutes, or at most a few hours, in most cases.
Penetration tests indicate that all of the material atomized by the impact burns immediately, so the "contaminant" from initial aerosols is uranium oxide. IF the sparks ignite other flammables that can heat the DU armor and the remaining bit of cold chisel, they will "burn" but the burning in this case is more like "extremely rapid rusting," and this "fire" goes out if external heat is removed. Solid hunks of DU over a few grams in size generally will not support combustion on their own.
Any DU that's burned will produce a small amount of vapor in addition to what is aerosolized by the impact, with similar distributions of particle sizes. Most of the bulk DU that's "burned" in a fire ends up as a lump of uranium oxide.
Ground samples at one site where ammunition of this kind was tested in large quantities indicate that the initial "spray" and fallout of heavier particles may produce DU concentrations equal to or exceeding background levels of Uranium common almost universally in soils taken at random, over a radius of about 20 meters. This remains a hazard and can be stirred up, tracked around, or eaten, but will not often be inhaled.
The aerosol bits small enough to be inhaled certainly aren't beneficial, but normal air circulation is likely to disperse them fairly rapidly. I haven't found precise figures on dispersal times, but in any reasonably breezy environment these small aerosols will be diluted to near background U levels before they get far from the impact area.
There is no question that there is contamination from these weapons, but it's mainly "chemical;" and there is no reason to scream "radioactive" because they simply aren't particularly dangerous from that aspect. The "radiation" in the immediate area may be somewhat above levels routinely encountered as background, but it's over a fairly small area. Most of it disperses and is diluted to background levels fairly quickly, and most of the rest settles out and could be cleaned up if entering the contaminated area or returning it to some public use is necessary. Most people, worldwide, eat more uranium in their turnips in a year than they are likely to get from any reasonably necessary exposure due to an incident in which these weapons and armor are used.
Other kinds of weapons uses for DU have been proposed and studied, and may be in use, but the rather scant information suggests that while they could cause somewhat larger areas of immediate contamination, the kind of hazard is unlikely to be very much different.
This is a composite opinion from 40+ separate laboratory reports on material properties, laboratory and manufacturing safety bulletins, individual shop safety standards, and repetitive (and rather boring) environmental impact surveys of a known areas of contamination some of which was extremely heavy. The basic material information is available, and based on what I've found, DU is a material that requires some degree of special handling and it poses some risks in use. It has a reasonably quantifiable health and environment impact; but it is not significantly more of a health or environmental hazard than many other fairly common materials - particularly among those used in warfare.