Aims. Hierarchical merging of galaxies plays an important role in galaxy formation and evolution. Mergers could trigger key evolutionary phases such as starburst activities and active accretion periods onto supermassive black holes at the centres of galaxies. We aim to detect mergers and merger stages (pre- and post-mergers) across cosmic history. Our main goal is to test whether it is more beneficial to detect mergers and their merger stages simultaneously or hierarchically. In addition, we wish to test the impact of merger time relative to the coalescence of merging galaxies. Methods. First, we generated realistic mock James Webb Space Telescope (JWST) images of simulated galaxies selected from the IllustrisTNG cosmological hydrodynamical simulations. The advantage of using simulations is that we have information on both whether a galaxy is a merger and its exact merger stage (i.e. when in the past or in the future the galaxy has experienced or will experience a merging event). Then, we trained deep-learning (DL) models for galaxy morphology classifications in the Zoobot Python package to classify galaxies into non-merging galaxies, merging galaxies and their merger stages. We used two different set-ups, a two-stage set-up versus a one-stage set-up. In the former set-up, we first classified galaxies into mergers and non-mergers, and we then classified the mergers into pre-mergers and post-mergers. In the latter set-up, non-mergers, pre-mergers and post-mergers were classified simultaneously. Results. We found that the one-stage classification set-up moderately outperforms the two-stage set-up. It offers a better overall accuracy and generally a better precision, particularly for the non-merger class. Out of the three classes, pre-mergers can be classified with the highest precision (∼65% versus ∼33% from a random classifier) in both set-ups, possibly because the merging features are generally more easily recognised, and because there are merging companions. More confusion is found between post-mergers and non-mergers than between these two classes and pre-mergers. The image signal-to-noise ratio (S/N) also affects the performance of the DL classifiers, but not by much after a certain threshold is crossed (S/N ∼ 20 in a 0.2″aperture). In terms of the merger timescale, both precision and recall of the classifiers strongly depend on merger time. Both set-ups find it more difficult to identify true mergers that are observed at stages that are farther from coalescence either in the past or in the future. For pre-mergers, we recommend selecting mergers that will merge in the next 0.4 Gyr to achieve a good balance between precision and recall.
