Let $G$ be a connected reductive complex algebraic group, and $E$ a complex elliptic curve. Let $G_E$ denote the connected component of the trivial bundle in the stack of semistable $G$-bundles on $E$. We introduce a complex analytic uniformization of $G_E$ by adjoint quotients of reductive subgroups of the loop group of $G$. This can be viewed as a nonabelian version of the classical complex analytic uniformization $E \simeq \mathbb{C}^*/q^{\mathbb{Z}}$. We similarly construct a complex analytic uniformization of $G$ itself via the exponential map, providing a nonabelian version of the standard isomorphism $\C^* \simeq \C/\Z$, and a complex analytic uniformization of $G_E$ generalizing the standard presentation $E = \C/(\Z \oplus \Z \tau )$. Finally, we apply these results to the study of sheaves with nilpotent singular support.