Coalition-based unmanned aerial vehicle (UAV) swarms have been widely used in urgent missions. To fasten the completion, mobile edge computing (MEC) has been introduced into UAV networks where coalition leaders act as servers to help members with data computing. This paper investigates a relative delay optimization in MEC-assisted UAV swarms. Considering that the scheduling methods have great impact on the delay, some theoretical analysis are made and a scheduling method based on shortest effective job first (SEJF) is proposed. Based on the coupled relationship between scheduling and resource allocation, the computation offloading and channel access problem are then jointly optimized. To solve the problem in distributed UAV networks, the optimization problem is formulated as an offloading game. It is proved that the game is an exact potential game (EPG) and it has at least one pure strategy Nash Equilibrium (PNE). To reach the PNE, a distributed offloading algorithm based on concurrent best-better response (CBBR) is designed. Finally, the simulations show that the performance of the proposed CBBR algorithm is better than traditional algorithms. Compared with other scheduling methods, the proposed scheduling method based on SEJF reduces the delay by up to 30%.