Despite superior rate-distortion performance over traditional codecs, Neural Image Compression (NIC) is limited by its computational scalability in practical deployment. Prevailing research focuses on accelerating specific NIC modules but is restricted in controlling overall computational complexity. To this end, this work introduces BaSIC (BayesNet structure learning for computational Scalable neural Image Compression), a comprehensive, computationally scalable framework that affords full control over NIC processes. We learn the Bayesian network (BayesNet) structure of NIC for controlling both neural network backbones and autoregressive units. The learning of BayesNet is achieved by solving two sub-problems, i.e., learning a heterogeneous bipartite BayesNet for the inter-node structure to regulate backbone complexity, and a multipartite BayesNet for the intra-node structure to optimize parallel computation in autoregressive units. Experiments demonstrate that our method not only facilitates full computational scalability with more accurate complexity control but also maintains competitive compression performance compared to other computation scalable frameworks under equivalent computational constraints. Code will be available after acceptance.