The proposed electron-proton collider experiments LHeC and FCC-eh at CERN are the highest resolution microscopes that can be realised in the present century and they would represent a really unique research facility. We exploit simulated neutral-current and charged-current deep-inelastic scattering data of the LHeC and the FCC-eh and examine their sensitivity for precision physics in the Electroweak sector of the Standard Model (SM), like the effective weak mixing angle $\sin^2\theta_{\text{W},\ell}^\text{eff}$, or the light-quark weak-neutral-current couplings. Unique measurements are further feasible at high precision for the running of the weak mixing angle, as well as for electroweak effects in charged current interactions. The sensitivity to beyond SM effects is studied using the generic $S$, $T$ and $U$ parameterization. The report summarizes previous studies about the LHeC and presents new prospects for the FCC-eh.