In the event of a tokamak disruption in a D-T plasma, fusion-born alpha particles take several milliseconds longer to thermalise than the background. As the damping rates drop drastically following the several orders of magnitudes drop of temperature, Toroidal Alfvén Eigenmodes (TAEs) can be driven by alpha particles in the collapsing plasma before the onset of the current quench. We employ kinetic simulations of the alpha particle distribution and show that the TAEs can reach sufficiently strong saturation amplitudes to cause significant core runaway electron (RE) transport in unmitigated ITER disruptions. As the eigenmodes do not extend to the plasma edge, this effect leads to an increase of the RE plateau current. Mitigation via massive material injection however changes the Alfvén frequency and can lead to mode suppression. A combination of the TAE-caused core RE transport with other perturbation sources could lead to a drop of runaway current in unmitigated disruptions.