Constructing earth-fixed cells with low-earth orbit (LEO) satellites in non-terrestrial networks (NTNs) has been the most promising paradigm to enable global coverage. The limited computing capabilities on LEO satellites however render tackling resource optimization within a short duration a critical challenge. Although the sufficient computing capabilities of the ground infrastructures can be utilized to assist the LEO satellite, different time-scale control cycles and coupling decisions between the space- and ground-segments still obstruct the joint optimization design for computing agents at different segments. To address the above challenges, in this paper, a multi-time-scale deep reinforcement learning (DRL) scheme is developed for achieving the radio resource optimization in NTNs, in which the LEO satellite and user equipment (UE) collaborate with each other to perform individual decision-making tasks with different control cycles. Specifically, the UE updates its policy toward improving value functions of both the satellite and UE, while the LEO satellite only performs finite-step rollout for decision-makings based on the reference decision trajectory provided by the UE. Most importantly, rigorous analysis to guarantee the performance convergence of the proposed scheme is provided. Comprehensive simulations are conducted to justify the effectiveness of the proposed scheme in balancing the transmission performance and computational complexity.