How is time-translation symmetry, or the lack of it, reflected in the Lagrangian formulation of mechanics? The answer is simple. In those cases where there is such symmetry, the Lagrangian has no explicit dependence on time. This is a subtle point: The value of the Lagrangian may vary with time, but only because the coordinates and velocities vary. Explicit time dependence means that the form of the Lagrangian depends on time.
