Onsetting the multi-messenger era with gravitational wave sources, the GW170817 event proved to be fundamental in addressing many open questions related to binary neutron star (BNS) mergers, demonstrating in particular that such physical systems can power short gamma-ray bursts (SGRBs). To understand the mechanisms behind the production of the corresponding relativistic jets, it is crucial to consistently model, via numerical simulations, the different phases involved, i.e. inspiral, merger, jet production, break-out, and further jet propagation. In this talk, I will present the first set of jet-launching BNS merger simulations performed with our general relativistic (GR) magnetohydrodynamic (MHD) code 'Spritz'. Focusing on the delayed-collapse scenario, in which the remnant neutron star survives for ≈25–50 ms after merger, for the first time we self-consistently capture two sequential outflows powered by distinct central engines—a pre-collapse remnant NS-collimated outflow and a post-collapse BH–disk jet—and their interaction. Next, I will introduce the procedure we developed to hand-off the outcome of BNS merger simulations into the special relativistic MHD code 'PLUTO' and show how the application of such a procedure to our jet-launching model allows us to consistently follow further jet evolution on much larger temporal and spatial scales. Finally, I will briefly introduce our new GPU-accelerated GRMHD code 'AsterX', aimed at high-resolution BNS merger simulations over larger temporal and spatial scales, with the goal of achieving predictive, end-to-end modelling of SGRB central engines.