We present results from our analysis of the Hydra I cluster observed in neutral atomic hydrogen (H I) as part of the Widefield ASKAP L-band Legacy All-sky Blind Survey (WALLABY). These WALLABY observations cover a 60-square-degree field of view with uniform sensitivity and a spatial resolution of 30 arcsec. We use these wide-field observations to investigate the effect of galaxy environment on H I gas removal and star formation quenching by comparing the properties of cluster, infall, and field galaxies extending up to ~5R200 from the cluster centre. We find a sharp decrease in the H I-detected fraction of infalling galaxies at a projected distance of ~1.5R200 from the cluster centre from $\sim 85{{\ \rm per\ cent}}$ to $\sim 35{{\ \rm per\ cent}}$. We see evidence for the environment removing gas from the outskirts of H I-detected cluster and infall galaxies through the decrease in the H I to r-band optical disc diameter ratio. These galaxies lie on the star-forming main sequence, indicating that gas removal is not yet affecting the inner star-forming discs and is limited to the galaxy outskirts. Although we do not detect galaxies undergoing galaxy-wide quenching, we do observe a reduction in recent star formation in the outer disc of cluster galaxies, which is likely due to the smaller gas reservoirs present beyond the optical radius in these galaxies. Stacking of H I non-detections with H I masses below $M_{\rm {HI}}\lesssim 10^{8.4}\, \rm {M}_{\odot }$ will be required to probe the H I of galaxies undergoing quenching at distances ≳60 Mpc with WALLABY.