This paper establishes tractable bounds of joint source-channel coding with hierarchical sources in the finite blocklength regime. In this setting, both the indirect source and observable source must be reconstructed under correlated distortion constraints, leading to a joint excess-distortion event. First, to build computable tight bounds, we introduce a novel $\mathsf{d}(\cdot)$-functional distortion relaxation, which enables tractable and tight bounding of the joint excess-distortion probability induced by correlated sources. By this approach, the nonasymptotic converse and achievability bounds are given. Second, Gaussian approximations for the proposed bounds are obtained, which are optimal for the transmission of a Gaussian memoryless source over an additive white Gaussian noise channel with mean-square error distortion. The optimal scheme is obtained via a structured analysis that captures the intrinsic tradeoff between semantic and observable reconstructions. Furthermore, for the transmission of Gaussian memoryless sources over AWGN channels, we obtain explicit and computable bounds, by providing a new geometric structure involving three correlated spherical regions. This results extend the classical two-spherical region analysis for a single distortion constraint. Numerical simulations demonstrate that the proposed achievability and converse bounds tightly sandwich the Gaussian approximation and align closely with Monte Carlo numerical results.