Large and electrically controllable spin splitting in nonmagnetic materials is highly desirable for condensed-matter physics and spintronics, enabling strongly spin-polarized quasiparticles and unconventional spin transport as well as their electrical control. Here, we establish a general framework for critical spin-polarized Weyl fermions in two-dimensional (2D) nonmagnetic systems. We predict the α-Bi-like Group-V puckered-layer family of 2D ferroelectrics as a material platform combining critical spin-Weyl fermions, giant spin splitting, and electrically switchable nonlinear and spin transport. Our calculations reveal exchange-scale spin splitting induced by spin-orbit coupling and a p-wave spin texture in the BiAs monolayer. Moreover, BiAs hosts four critical spin-Weyl points, each formed within a single-spin channel. The nearly quantized ⟨ S_z⟩ is protected by an emergent S_z-conservation symmetry in the low-energy limit. The spin-Weyl cones also give rise to unconventional spin-transport phenomena, including a purely out-of-plane Edelstein response and a pronounced spin Nernst effect. Ferroelectric reversal further enables electrical switching of the p-wave spin texture and inversion-odd responses, while the tunability of these properties through composition, strain, and magnetic field is also systematically demonstrated across the entire family. These results establish both a general nonmagnetic framework and a promising material platform for exploring critical spin-Weyl physics, nonlinear Hall phenomena, and unconventional spin transport.