The complete gauge-invariant set of the one-loop QED corrections to the parity-nonconserving (PNC) amplitude in cesium and francium is evaluated to all orders in αZ using a local form of the Dirac-Fock potential. The calculations are performed in both length and velocity gauges for the absorbed photon and the total binding QED correction is found to be -0.27(3)% for Cs and -0.28(5)% for Fr. Moreover, a high-precision calculation of the electron-correlation and Breit-interaction effects on the 7s-8s PNC amplitude in francium using a large-scale configuration-interaction Dirac-Fock method is performed. The obtained results are employed to improve the theoretical predictions for the PNC transition amplitude in Cs and Fr. Using an average value from two most accurate measurements of the vector transition polarizability, the weak charge of Cs133 is derived to amount to QW=-72.65(29)exp(36)theor. This value deviates by 1.1σ from the prediction of the standard model. The values of the 7s-8s PNC amplitude in Fr223 and Fr210 are obtained to be -15.49(15) and -14.16(14), respectively, in units of i×10-11(-QW) a.u.