Observations of the time lag between the last deglaciation and a surge in volcanic activity in Iceland constrain the average melt ascent velocity to be $\ge 50$ $m/\mathrm{yr}$. Although existing theoretical work has explained why the surge in eruption rates increased $5$-$30$ fold from the steady-state rates during the last deglaciation, they cannot account for large variations of Rare Earth Element (REE) concentrations in the Icelandic lavas. Lavas erupted during the last deglaciation are depleted in REEs by up to $70\%$; whereas, existing models, which assume instantaneous melt transport, can only produce at most $20\%$ depletion. Here, we develop a numerical model with finite melt ascent velocity and show that the variations of REEs are strongly dependent on the melt ascent velocity. When the average melt ascent velocity is $100$ $m/\mathrm{yr}$, the variation of $\mathrm{La}$ calculated by our model is comparable to that of the observations. In contrast, when the melt ascent velocity is $1,000$ $m/\mathrm{yr}$ or above, the model variation of $\mathrm{La}$ becomes significantly lower than observed, which explains why previous models with instantaneous melt transport did not reproduce the large variations. We provide the first model that takes account of the diachronous response of volcanism to deglaciation. We show by comparing our model calculations of the relative volumes of different eruption types (subglacial, finiglacial and postglacial) and the timing of the bursts in volcanic eruptions with the observations across different volcanic zones that the Icelandic average melt ascent velocity during the last deglaciation is likely to be $\sim 100$ $m/\mathrm{yr}$.