Computer simulations are invaluable for the study of ultrafast phenomena, as it is not possible to directly access the electronic and nuclear dynamics in experiments. We present an efficient method for simulating the time-dependent coupled electron-ion dynamics within the Ehrenfest picture in molecules under the influence of time-dependent electric fields, based on an extension of the density-functional tight binding model. We consider self-consistency in a self-multipole-consistent framework, expanding the electron density in terms of radial Gaussian and angular real cubic harmonic functions. This enables the efficient computation of the electrostatic interaction energy while retaining a physical description of charge transfer and ionic polarisation. We show that this Gaussian tight binding method produces molecular polarisabilities, time-dependent dipole moments, and electron densities in strong agreement with density-functional theory, but at a small fraction of the cost. This efficiency enables high-throughput ultrafast studies on molecules, which we demonstrate on the example of transient core-spectroscopy on polythiophene fragments.