Non-terrestrial Networks (NTN) will play a key role in enabling a fully connected, digitized, and intelligent society through their integration into traditional terrestrial communication networks in the upcoming 6G world. The space-based Orbital Edge Computing (OEC) has already shown some promising results in terms of boosting the capacity, coverage, security, and resilience of communication systems. Among others, Low Earth Orbit (LEO) satellite constellations can be beneficial in terms of reduced communication distances and easy and flexible deployments. With integrated OEC facilities into NTN platforms, a new era of space computing has begun. It has several benefits in terms of serving space/ground users through satellite computation, communication, and storage resources. However, with size limitations, each LEO satellite node can have a limited amount of resources. This restricts the number of services provided by the satellites and the users served. Therefore a proper user-satellite assignment is needed. Additionally, with a boost in communication technologies, different terrestrial and NTN layers can form multi-tier computing facilities with higher capacity and coverage. We aim to solve the network selection problem over multi-service multi-tier edge computing facilities provided by multiple LEO constellations and cloud computing facilities. A Hierarchical Reinforcement Learning (HRL) based solution is proposed for optimizing the multi-level network selection decisions to minimize the task processing latency. The simulation results show improvements in terms of latency performance and service reliability, considering user/system constraint satisfaction.