Uranium-238, the most prevalent isotope in uranium ore, has a half-life of about 4.5 billion years; that is, half the atoms in any sample will decay in that amount of time. Uranium-238 decays by alpha emission into thorium-234, which itself decays by beta emission to protactinium-234, which decays by beta emission to uranium-234, and so on. The various decay products, (sometimes referred to as "progeny" or "daughters") form a series starting at uranium-238. After several more alpha and beta decays, the series ends with the stable isotope lead-206.
Uranium-238 emits alpha particles which are less penetrating than other forms of radiation, and weak gamma rays. As long as it remains outside the body, uranium poses little health hazard (mainly from the gamma-rays). If inhaled or ingested, however, its radioactivity poses increased risks of lung cancer and bone cancer. Uranium is also chemically toxic at high concentrations and can cause damage to internal organs, notably the kidneys. Animal studies suggest that uranium may affect reproduction, the developing fetus,(1) and increase the risk of leukemia and soft tissue cancers.(2)
While U-235 has a half-life of 704 million years, because of the 0.7% U-235 prevalence in natural uranium, most of the decays in natural uranium come from U-238 (and an equivalent amount from the daughter products, such as U-234, thorium, protactinium, etc., which all decay more rapidly, sometimes much more). Depleted uranium (>> 99% U-238) is not much different WRT this decay chain.
It is true that U-238 emits alphas (and weak gamma). Alphas are the most easilty stopped; a piece of paper will do it. The problem comes when it's inside the body; when that happens, what stops it is your body tissues! And the vapourised uranium from munitions produces lots of neat little particles that get all through your body....