We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely Helly spaces, and strongly shortcut spaces. We show that any hierarchically hyperbolic space admits a new metric that is coarsely Helly. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that any coarsely Helly metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups---including mapping class groups of surfaces---are coarsely Helly and coarsely Helly groups are strongly shortcut. Using these results we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, are of type $FP_{\infty}$, have finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.