A formula for computing approximate leakage of population from an initially prepared electronic state with a nonequilibrium nuclear distribution to a second nonadiabatically coupled electronic state is derived and applied. The formula is a nonequilibrium generalization of the familiar golden rule, which applies when the initial nuclear state is a rovibrational eigenstate of the potential energy surface associated with the initially populated electronic state. Here, more general initial nuclear states are considered. The resultant prescription, termed the nonequilibrium golden rule formula, can be evaluated via semiclassical procedures and hence applied to multidimensional, e.g., condensed phase systems. To illustrate its accuracy, application is made to a spin-boson model of "inner sphere" electron transfer. This model, introduced by Garg et al. [J. Chem. Phys. 83, 4491 (1985)] for the nonadiabatic transition out of a thermal distribution of states in the initial (donor) electronic level, is extended to include nonequilibrium, nonstationary initial nuclear states on the donor surface. The predictions of the nonequilibrium golden rule are found to agree well with numerically exact path integral results for a wide range of initial distortions of the initial nuclear wave packet from its equilibrium configuration.