Holistic co-speech motion generation aims to synthesize coordinated facial expressions, hand gestures, and body movements from speech, which is challenging due to heterogeneous part-specific dynamics, long-range temporal coherence, and cross-part coordination. Directly modeling high-dimensional motion can be unstable, while overly compact or unstructured latent spaces may lose fine-grained motion details. We present \textbf{TalkFlow}, a structured latent flow framework for speech-driven holistic motion generation. TalkFlow learns a continuous three-stream latent representation for upper-body motion, lower-body motion, and facial expressions, preserving part-specific dynamics without forcing heterogeneous motion components into a single latent space. To maintain holistic coordination, lightweight cross-stream interactions are incorporated in both latent representation learning and flow-based generation. Conditioned on multi-level speech features, TalkFlow performs speech-conditioned latent flow matching, learning a continuous transport process from source latents to realistic motion latents. Experiments on the BEAT2 dataset demonstrate strong performance across facial expression accuracy, body gesture quality, and speech-motion synchronization. Qualitative comparisons and user studies further show that TalkFlow generates more natural, expressive, and speech-aligned holistic motion than state-of-the-art methods. The project page is available at https://xinmuwang.github.io/TalkFlow/.

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