Depleted uranium in tanks and ammunition.?

No. Unless you eat some of it. The primary radiation danger from depleted uranium is due to alpha particles, which do not travel far through air, and do not penetrate clothing or the outer layers of skin. And if you slap a coat of paint on top of it guess what? Thus, the primary concern is internal exposure, due to inhalation, ingestion or shrapnel contamination.

Considering the precautions the army takes to protect a soldier I would find it very hard to believe they would let soldiers near unprotected uranium.

Think about basic and your training they made triply sure all equipment was fit for use.

Depleted uranium only gives off alpha rays at such a low level that your skin protects you from it.

Only if they eat the ammunition for lunch, which seems unlikely. Otherwise, no.