This brief aims at the issue of globally composite-learning-based intelligent fast finite-time (F-FnT) tracking control for a class of uncertain systems in strict-feedback form subject to nonlinearly periodic disturbances. First, uncertain dynamics with periodic parameters are identified by incorporating Fourier series expansion (FSE) into intelligent estimator, which leverages the feedback of newly designed prediction errors in updating weights to boost learning performance. Then, a novel switching mechanism is constructed to fulfill smoothly switch from the compound FSE-based intelligent controller to robust control law when the inputs of intelligent estimator transcend the valid approximation domain. By fusing the switching mechanism with an improved F-FnT backstepping algorithm, the globally F-FnT boundedness of all variables in the closed-loop system is guaranteed. Finally, a simulation study is conducted to evince the availability of the mathematical results.