A time-dependent nonlinear theory for gyroklystron amplifier is presented. The theory includes a time-dependent description of the electromagnetic fields and a self-consistent analysis of the electrons. The generalized telegrapher equations represent the electromagnetic fields. The equations of motion of the electrons are described in the framework of the guiding-center approximation. All trajectories are calculated and used as current sources for the fields. The nonlinear theory of interaction is investigated in which mode coupling is taken into account in varying wall radius. Transverse velocity of the electrons from the gyroklystron amplifier satisfies Gaussian distribution. Distribution model of the velocity spread in the gyroklystron amplifier beam-wave interaction is established. A code for the self-consistent nonlinear beam-wave interaction is developed based on the presented theory. The electron beam-wave interaction of a Kα band gyroklystron amplifier is thoroughly studied and analyzed by the code. Numerical verification using MAGIC simulation is also given. The numerical results are in good agreement with the self-consistent nonlinear simulations.