In this paper, we propose a novel cooperative time synchronization (CTS) scheme via broadcasting for time-sensitive cell-free (CF) massive multiple-input multiple-output (MIMO) networks, by allowing distributed access points (APs) to jointly broadcast synchronization messages to the target nodes. With the proposed CTS scheme, more timestamps from distributed APs are available to achieve higher synchronization accuracy over the conventional non-cooperative time synchronization approaches. Parameter estimation is conducted by maximum likelihood estimation of clock offset and clock skew under the Gaussian transmission delay model, and the corresponding Cramer-Rao lower bounds (CRLBs) are derived. In addition, the clock offset and skew deviations among APs are also estimated, which are fed back to APs for further clock adjustment. Thanks to the AP association scheme and broadcast characteristics, the proposed CTS scheme demonstrates robustness against clock parameter deviations, while at a relatively low overhead and computational complexity.