The magnetization reversal in ordered arrays of iron oxide nanotubes of 50 nm outer diameter grown by atomic layer deposition is investigated theoretically as a function of the tube wall thickness $d_w$. In thin tubes $\left(d_w<13\text{nm}\right)$ the reversal of magnetization is achieved by the propagation of a vortex domain boundary, while in thick tubes $\left(d_w>13\text{nm}\right)$ the reversal is driven by the propagation of a transverse domain boundary. Magnetostatic interactions between the tubes are responsible for a decrease in the coercive field in the array. Our calculations are in agreement with recently reported experimental results. We predict that the crossover between the vortex and transverse modes of magnetization reversal is a general phenomenon on the length scale considered.