The magnetohydrodynamics of unsteady magnetic merging flows in one space dimension are investigated by finite difference methods. The evolutions of the x magnetic field, y velocity, z electric current, pressure, and density are followed as functions of y and t in the presence of finite conductivity. For an initial state consisting of a neutral sheet current layer in static mechanical equilibrium the computations indicate significant compressible MHD evolution in times comparable to the Alfven crossing time. Rapid inflow begins behind large-amplitude magnetosonic expansion waves radiated from the current layer. For the magnetic Reynolds numbers studied (Rm up to 100), this one-dimensional evolution occurs rapidly compared with the growth time of the tearing mode instability and may have an important effect on its development. The magnitude of the current density near the neutral sheet was found to decrease steadily with time in all cases with static initial states in mechanical equilibrium. This may pose a difficulty for models which invoke an enhancement of the current density there in order to excite plasma microinstabilities that yield a large anomalous resistivity and rapid magnetic energy release. The lack of significant current density enhancement can also be seen in previous two-dimensional simulations which do not invoke large externally imposed inflow velocities. Only with externally imposed stresses do current density enhancements appear in the present one-dimensional simulations. The effects of varying Rm and the plasma adiabatic index, as well as the initial conditions, are investigated.