Unmanned Aerial Vehicle (UAV)-assisted Low Earth Orbit (LEO) satellite edge computing (ULSE) networks can address the challenge communications issues in areas with harsh terrain and achieve global wireless coverage to provide services for mobile user devices (MUDs). This paper studies the LEO-UAV task offloading problem where MUDs compete for limited resources in the ULSE networks. We formulate the optimization problem with the goal of minimizing the cost of all MUDs while meeting resource constraint and satellite coverage time constraint. We first theoretically prove that this problem is NP-hard. We then reformulate the problem as a LEO-UAV task offloading game (LUTO-Game), and show that there is at least one Nash equilibrium solution for the LUTO-Game. We propose a joint UAV and LEO satellite task offloading (JULTO) algorithm to obtain the Nash equilibrium offloading strategy, and analyze the performance of the worst-case offloading strategy obtained by the JULTO algorithm. Finally, extensive experiments, including convergence analysis and comparison experiments, are carried out to validate the effectiveness of our JULTO algorithm.