Achieving seamless viewing of high-fidelity volumetric video, comparable to 2D video experiences, remains an open challenge. Existing volumetric video compression methods either lack the flexibility to adjust quality and bitrate within a single model for efficient streaming across diverse networks and devices, or struggle with real-time decoding and rendering on lightweight mobile platforms. To address these challenges, we introduce 4DGCPro, a novel hierarchical 4D Gaussian compression framework that facilitates real-time mobile decoding and high-quality rendering via progressive volumetric video streaming in a single bitstream. Specifically, we propose a perceptually-weighted and compression-friendly hierarchical 4D Gaussian representation with motion-aware adaptive grouping to reduce temporal redundancy, preserve coherence, and enable scalable multi-level detail streaming. Furthermore, we present an end-to-end entropy-optimized training scheme, which incorporates layer-wise rate-distortion (RD) supervision and attribute-specific entropy modeling for efficient bitstream generation. Extensive experiments show that 4DGCPro enables flexible quality and multiple bitrate within a single model, achieving real-time decoding and rendering on mobile devices while outperforming existing methods in RD performance across multiple datasets. Project Page: https://mediax-sjtu.github.io/4DGCPro