When a current flows through a wire having a nonuniform...

By the principle of conservation of charge, the current $I$ (which is the amount of charge crossing a given cross-section in a given time interval) remains constant throughout the wire in a steady state, regardless of its cross-sectional area.

The free-electron density $n$ is a characteristic property of the material of the wire and is independent of its dimensions.

The current density is given by $J = \dfrac{I}{A}$, which is inversely proportional to the cross-sectional area $A$.

The drift speed is given by $v_d = \dfrac{I}{neA}$, which is also inversely proportional to the cross-sectional area $A$.

Thus, the amount of charge crossing in a given time interval and the free-electron density are independent of the cross-sectional area.