Cell-free massive MIMO (CF mMIMO) is a promising next generation wireless architecture to realize federated learning (FL). However, sensitive information of user equipments (UEs) may be exposed to the involved access points or the central processing unit in practice. To guarantee data privacy, effective privacy-preserving mechanisms are defined in this paper. In particular, we demonstrate and characterize the possibility in exploiting the inherent quantization error, caused by low-resolution analog-to-digital converters (ADCs) and digital-to-analog converters (DACs), for privacy-preserving in a FL CF mMIMO system. Furthermore, to reduce the required uplink training time in such a system, a stochastic non-convex design problem that jointly optimizing the transmit power and the data rate is formulated. To address the problem at hand, we propose a novel power control method by utilizing the successive convex approximation approach to obtain a suboptimal solution. Besides, an asynchronous protocol is established for mitigating the straggler effect to facilitate FL. Numerical results show that compared with the conventional full power transmission, adopting the proposed power control method can effectively reduce the uplink training time under various practical system settings. Also, our results unveil that our proposed asynchronous approach can reduce the waiting time at the central processing unit for receiving all user information, as there are no stragglers that requires a long time to report their local updates.