A search for new heavy vector resonances decaying into a $W$ boson and a Standard-Model Higgs boson ($h$), targeting a semileptonic final state where the $W$ boson decays into a lepton-neutrino pair and the Higgs boson decays into a b-quark pair, is performed using 139 fb\textsuperscript{-1} of $\sqrt{s}$ = 13 TeV proton-proton collision data collected by the ATLAS detector at the Large Hadron Collider (LHC) during LHC Run 2 (2015-2018). The search probes a wide range of potential heavy resonance masses, from 400 GeV to 5 TeV, by examining the invariant and transverse mass distributions of $Wh$ candidates for a localized excess. Many beyond Standard Model (BSM) theories generically predict such diboson vector resonances, such as the Heavy Vector Triplet (HVT) benchmark model through which the results are interpreted. No significant excess over Standard Model predictions is observed, and 95\% confidence level upper limits are placed on the production cross-section times branching ratio. These limits are also converted into constraints on the parameter space of the HVT model. Furthermore, reconstructing the trajectories of charged particles through the ATLAS Inner Detector is a critical component of both this search and the broader ATLAS physics program, but represents an enormous computational challenge due to its inherent combinatorial complexity. This work also presents successful efforts to improve and optimize this process, in both computational resources and physics performance, for LHC Run 3 (2022-2025). Under design accelerator conditions, the improved ATLAS track reconstruction is twice as fast as the legacy version, with no significant reduction in overall efficiency and a more than two-fold reduction in ``fake" tracks.