This paper introduces a new market-based distributed algorithm to solve a task assignment problem arising in missions using multiple unmanned aerial vehicles with timing coordination. The proposed algorithm is composed of four phases analogous to the activities of participants in a market. Analyses on the convergence and scalability of the proposed algorithm indicate that the algorithm has a polynomial-time complexity, which is verified through numerical experiments as well. The proposed algorithm is applied for a suppression of enemy air defenses mission with pop-up threats and loss of unmanned aerial vehicles. Numerical simulations demonstrate that the proposed algorithm operates in a robust way under a dynamic environment and can be potentially considered for actual cooperative timing missions for unmanned aerial vehicles.