The spectacular head-on collision of the two gas-rich galaxies of the Taffy system, UGC 12914/15, gives us a unique opportunity to study the consequences of a direct ISM-ISM collision. To interpret existing multi-wavelength observations, we made dynamical simulations of the Taffy system including a sticky particle component. To compare simulation snapshots to HI and CO observations, we assume that the molecular fraction of the gas depends on the square root of the gas volume density. For the comparison of our simulations with observations of polarized radio continuum emission, we calculated the evolution of the 3D large-scale magnetic field for our simulations. The induction equations including the time-dependent gas-velocity fields from the dynamical model were solved for this purpose. Our simulations reproduce the stellar distribution of the primary galaxy, UGC 12914, the prominent HI and CO gas bridge, the offset between the CO and HI emission in the bridge, the bridge isovelocity vectors parallel to the bridge, the HI double-line profiles in the bridge region, the large line-widths (~200 km/s) in the bridge region, the high field strength of the bridge large-scale regular magnetic field, the projected magnetic field vectors parallel to the bridge and the strong total power radio continuum emission from the bridge. The stellar distribution of the secondary model galaxy is more perturbed than observed. The observed distortion of the HI envelope of the Taffy system is not reproduced by our simulations which use initially symmetric gas disks. The model allows us to define the bridge region in three dimensions. We estimate the total bridge gas mass (HI, warm and cold H2) to be 5 to 6 10^9 M_sun, with a molecular fraction M_H2/M_HI of about unity (abrigded).