We consider a control constrained nonlinear optimal control problem under perturbations represented by a parameter which is a function of time. Our main assumptions involve smoothness of the functions appearing in the integral functional and the state equations, an integral coercivity condition, and a condition that the reference optimal control is an isolated solution of the variational inequality for the control appearing in the maximum principle. We also consider a corresponding discrete-time optimal control problem obtained from the continuous-time one by applying the Euler finite-difference scheme. Based on an enhanced version of Robinson's implicit function theorem, we establish that there exists a natural number (N) over bar such that if the number N of the grid points is greater than (N) over bar, then the solution mapping of the discrete-time problem has a Lipschitz continuous single-valued localization with respect to the parameter, whose Lipschitz constant and the sizes of the neighborhoods involved do not depend on N. As an application, we show that the Newton/SQP method converges uniformly with respect to the step-size of the discretization and small changes of the parameter. Numerical experiments with a satellite optimal control problem illustrate the convergence result.