We explore the joint detection prospects of short gamma-ray bursts (sGRBs) and their gravitational wave (GW) counterparts by the current and upcoming high-energy GRB and GW facilities from binary neutron star (BNS) mergers. We consider two GW detector networks: (1) a four-detector network comprising LIGO Hanford, Livingston, Virgo, and Kagra (IGWN4) and (2) a future five-detector network including the same four detectors and LIGO India (IGWN5). For the sGRB detection, we consider existing satellites Fermi and Swift and the proposed all-sky satellite Daksha. Most of the events for the joint detection will be off-axis, hence, we consider a broad range of sGRB jet models predicting the off-axis emission. Also, to test the effect of the assumed sGRB luminosity function, we consider two different functions for one of the emission models. We find that for the different jet models, the joint sGRB and GW detection rates for Fermi and Swift with IGWN4 (IGWN5) lie within 0.07–0.62  yr⁻¹ 0.8–4.0 yr⁻¹) and 0.02–0.14 yr⁻¹ (0.15–1.0 yr⁻¹), respectively, when the BNS merger rate is taken to be 320 Gpc⁻³ yr⁻¹. With Daksha, the rates increase to 0.2–1.3 yr⁻¹ (1.3–8.3 yr⁻¹), which is 2–9 times higher than the existing satellites. We show that such a mission with higher sensitivity will be ideal for detecting a higher number of fainter events observed off-axis or at a larger distance. Thus, Daksha will boost the joint detections of sGRB and GW, especially for the off-axis events. Finally, we find that our detection rates with optimal SNRs are conservative, and noise in GW detectors can increase the rates further.