The postprocesses that occur in coelectrospun polymer nanofibers are investigated. The high rate of solvent evaporation during the electrospinning of fibers results in such a rapid formation of the shell of the tubular nanofibers, that the polymer molecules composing the fibers are in a nonequilibrium state. This stretched state of macromolecules is assumed to be stabilized in a solid matrix, and can account for the anomalous properties of the nanofibers. During this processing stage a considerable amount of solvent remains inside the tubular nanofibers. The evaporation of the solvent continues for several minutes, and is accompanied by further evolution of the nanofibers in both their microstates and macrostates. In this paper, we examine possible macrostate modifications of the nanofibers, such as radial buckling. The theoretical model which describes the kinetics of the solvent evaporation is found to be in good agreement with experimental observations. Thus, the physical parameters of the system in question can be estimated, and the conditions of fiber shell instability that produce buckling of the tubular nanofibers can also be predicted.