In this work, we study magnetization switching induced by spin-orbit torque in W(Pt)/Co/NiO heterostructures with variable thickness of heavy-metal layers W and Pt, perpendicularly magnetized Co layer and an antiferromagnetic NiO layer. Using current-driven switching, magnetoresistance and anomalous Hall effect measurements, perpendicular and in-plane exchange bias field were determined. Several Hall-bar devices possessing in-plane exchange bias from both systems were selected and analyzed in relation to our analytical switching model of critical current density as a function of Pt and W thickness, resulting in estimation of effective spin Hall angle and perpendicular effective magnetic anisotropy. We demonstrate in both the Pt/Co/NiO and the W/Co/NiO systems the deterministic Co magnetization switching without external magnetic field which was replaced by in-plane exchange bias field. Moreover, we show that due to a higher effective spin Hall angle in W than in Pt-systems the relative difference between the resistance states in the magnetization current switching to difference between the resistance states in magnetic field switching determined by anomalous Hall effect ($Delta R/Delta R_{text{AHE}}$) is about twice higher in W than Pt, while critical switching current density in W is one order lower than in Pt-devices. The current switching stability and training process is discussed in detail.