Monotone chain polar codes generalize classical polar codes to multivariate settings, offering a flexible approach for achieving the entire admissible rate region in the distributed lossless coding problem. However, this flexibility also introduces significant challenges for existing successive cancellation (SC) based decoding schemes. Motivated by the need for a general SC decoding solution, we present a comprehensive decoding strategy for monotone chain polar codes that can handle arbitrary numbers of terminals, non-binary alphabets, and decoding along arbitrary monotone chains. Specifically, we formulate the SC decoding task as a series of inference subtasks over the polar transform and propose a computational graph framework based on probability propagation principles. This approach highlights the impact of variable switching during decoding and shows that time complexity varies between $O(N\log{N})$ and $O(N^2)$, depending on the specific chain structure. Moreover, we demonstrate that the widely used $O(N)$ space optimization is not universally applicable to monotone chain polar codes, which prompts us to introduce a constant-time decoder forking strategy based on the proposed logical computation graphs. This strategy enables time-efficient list decoding without relying on $O(N)$-space techniques. Numerical results verify the superior performance of the proposed scheme compared with the classical lazy-copy scheme.