Using time-dependent Ginzburg-Landau theory, we study the role of amplitude and phase fluctuations in the recovery of charge and spin stripe phases in response to a pump pulse that melts the orders. For parameters relevant to the case where charge order precedes spin order thermodynamically, amplitude recovery governs the initial time scales, while phase recovery controls behavior at longer times. In addition to these intrinsic effects, there is a longer spin re-orientation time scale related to the scattering geometry that dominates the recovery of the spin phase. Coupling between the charge and spin orders locks the amplitude and similarly the phase recovery, reducing the number of distinct time scales. Our results well reproduce the major experimental features of pump-probe x-ray diffraction measurements on the striped nickelate La$_{1.75}$Sr$_{0.25}$NiO$_4$. They highlight the main idea of this work, which is the use of time-dependent Ginzburg-Landau theory to study systems with multiple coexisting order parameters.