We extend the time-dependent Gutzwiller variational approach, recently introduced by Schirò and Fabrizio [Phys. Rev. Lett. 105, 076401 (2010)], to impurity problems. Furthermore, we derive a consistent theory for the steady state, and show its equivalence with the previously introduced nonequilibrium steady-state extension of the Gutzwiller approach. The method is shown to be able to capture dissipation in the leads, so that a steady state is reached after a sufficiently long relaxation time. The time-dependent method is applied to the single-orbital Anderson impurity model at half filling, modeling a quantum dot coupled to two leads. In these exploratory calculations, the Gutzwiller projector is limited to act only on the impurity. The strengths and the limitations of this approximation are assessed via comparison with state-of-the-art continuous-time quantum Monte Carlo results. Finally, we discuss how the method can be systematically improved by extending the region of action of the Gutzwiller projector.