This paper is interested in joint T₁ - T₂ inversion from borehole nuclear magnetic resonance (NMR) measurements when a limited number of wait times (WTs) are used. Unlike a straightforward representation of the multi-WT NMR measurements over an overcomplete kernel matrix and using a sparsity-aware inversion method, the paper proposes to exploit the two-dimensional sparsity in the T₁ - T₂ domain and the smoothness (i.e., the local cluster feature) simultaneously by using a sparse clustered signal representation. To enable a fully automated workflow for borehole NMR logging operations, a sparse clustered Bayesian-inspired algorithm is developed to exploit the sparsity in the solution space while learning the noise statistics. It in turn mitigates a cumbersome issue of selecting regularization parameters in deterministic sparsity-aware methods and reduces the human intervention. Several enhancements are discussed to reduce the computational complexity and improve the robustness at low signal-to-noise ratios. The proposed algorithm has been tested on synthetic datasets and lab samples and the results confirm its effectiveness.