Partial melting of the asthenospheric mantle generates the magma that supplies volcanic systems. The timescale of melt extraction from the mantle has been hotly debated. Microstructural measurements of the permeability of partially molten rocks typically suggest relatively slow melt extraction (1 m/yr) [e.g. 1]. By contrast, inferences from geochemical measurements of Uranium series and geophysical observations typically point to much faster melt extraction (100 m/yr) [e.g. 2]. The most recent deglaciation of Iceland caused the mantle below to depressurise, triggering additional mantle melting and magma flux at the surface, which has been extensively mapped. The rapid response of magmatic activity to deglaciation has been used to argue for relatively rapid melt extraction [3,4]. Perhaps, however, this unusual period when magma fluxes increased several-fold is not representative of steady-state melt velocities under Iceland, let alone the mid-ocean ridge system more generally. We develop a one-dimensional, but time-dependent and fully nonlinear, model of the generation and transport of mantle melts force by time-dependent ice unloading. We show that these models are sensitive to the nonlinear nature of the system, namely that the melt velocities are faster during and following a deglaciation event. For a given nonlinear model, we show that an equivalent linear estimate of the steady-state melt velocity is too fast. We calculate an overestimation factor as a function of the factor of mantle melting caused by deglaciation. For the most recent, and best mapped, deglaciation, we show that about 30 m/yr is the best estimate of melt velocity. This is a factor of 3 smaller than previously claimed [4], but still relatively fast. Finally, we discuss the applicability of these results to the mid-ocean ridge system by considering the role of spreading rate and the plume-influence on Iceland. [1] Wark, D. et al. (2003). JGR. doi:10.1029/2001JB001575 [2] Stracke, A., Bourdon, B., & McKenzie, D. (2006). EPSL. doi:10.1016/j.epsl.2006.01.057 [3] Maclennan, J. et al. (2002). Geochem. Geophys. Geosyst. doi:10.1029/2001GC000282 [4] Eksinchol I., Rudge J.F., Maclennan J. (2019) Geochem. Geophys. Geosyst. doi:10.1029/2019GC008222