We consider the use of the polynomial chaos method to approximate optimal control problems with randomly varying uncertain parameters by deterministic surrogate problems. Our focus is on nonlinear problems that require a high expansion order to give meaningful statements about the uncertainty propagation. Their resulting size and complexity pose a computational challenge for traditional optimal control methods. Our contributions include an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision, as is highlighted in a numerical case study with two nonlinear real-world problems. The algorithmic contribution is complemented by a convergence proof showing that the optimal control solutions approach the correct solution for increasing expansion orders.