Binary neutron star (BNS) mergers are crucial sources for multi-messenger astrophysics, linking gravitational waves (GWs), short gamma-ray bursts (sGRBs), and potentially very-high-energy (VHE) gamma-ray and neutrino emission. In this work, we focus on VHE GRB~160821B, one of the nearest known sGRBs, to investigate the prospects for joint detection by the LIGO/Virgo/KAGRA (LVK) network and the upcoming Cherenkov Telescope Array (CTA). Although GRB~160821B lies beyond the expected GW detection horizon at its actual distance, we explore hypothetical scenarios placing it within the O5 BNS horizon, employing a structured Gaussian jet model to assess how afterglow microphysical parameters strongly influence synchrotron self-Compton (SSC) detectability by CTA and the likelihood of simultaneous GW detection. Additionally, we estimate the high-energy neutrino fluxes from GRB~160821B, finding them to lie below current detection thresholds, but found that smaller luminosity distance or higher $\gamma$-ray efficiency could substantially improve their detectability. Our study delineates the parameter space favourable for joint GW–VHE observations and underscores the multi-messenger potential of nearby sGRBs.