The performance of $L$-branch scan-and-wait combining (SWC) reception systems over arbitrarily correlated and not necessarily identically distributed Nakagami-$m$ fading channels is analyzed and evaluated. Firstly, a fast convergent infinite series representation for the SWC output signal-noise ratio (SNR) is presented. This expression is used to obtain analytical expressions in the form of infinite series for the average error probability performance of various modulation schemes for integer values of $m$ as well as the average number of paths estimation and average waiting time (AWT) of $L$-branch SWC receivers for arbitrary values of $m$. The numerically obtained results have shown that the performance expressions converge very fast to their exact analytical values. It was found that the convergence speed depends on the correlation and operating SNR values as well as the Nakagami $m$-parameter. In addition to the analytical results, complementary computer simulated performance evaluation results have been obtained by means of Monte Carlo error counting techniques. The match between these two sets of results has verified the accuracy of the proposed mathematical analysis. Furthermore, it is revealed that, at the expense of a negligible AWT, the average error probability performance of SWC receivers is always superior to that of switched-and-examine combining receivers and in certain cases to that of maximal-ratio combining receivers.