In the current work, we have implemented an extension of the standard Gaussian Process formalism, namely the Multi-Task Gaussian Process with the ability to perform a joint learning of several cosmological data simultaneously. We have utilised the low-redshift expansion rate data from Supernovae Type-Ia (SN), Baryon Acoustic Oscillations (BAO) and Cosmic Chronometers (CC) data in a joint analysis. We have tested several possible models of covariance functions and find very consistent estimates for cosmologically relevant parameters. In the current formalism, we also find provisions for heuristic arguments which allow us to select the best-suited kernel for the reconstruction of expansion rate data. We also utilised our method to account for systematics in CC data and find an estimate of $H_0 = 68.52^{+0.94 + 2.51 (sys)}_{-0.94} $ $textrm{km/s Mpc}^{-1}$ and a corresponding $r_d = 145.61^{+2.82}_{ - 2.82 - 4.3 (sys)} $ Mpc as our primary result. Subsequently, we find constraints on the present deceleration parameter $q_0 = -0.52 pm 0.06$ and the transition redshift $z_T = 0.64^{+0.12}_{-0.09}$. All the estimated cosmological parameters are found to be in good agreement with the standard $Lambda$CDM scenario. Including the local model-independent $H_0$ estimate to the analysis we find $H_0 = 71.40^{ + 0.30 + 1.65 (sys)}_{- 0.30 } $ $textrm{km/s Mpc}^{-1}$ and the corresponding $r_d = 141.29^{ + 1.31 }_{-1.31-2.63 (sys)}$ Mpc. Also, the constraints on $r_d H_0$ remain consistent throughout our analysis and also with the model-dependent CMB estimate. Using the $mathcal{O}m(z)$ diagnostic, we find that the concordance model is very consistent within the redshift range $z lesssim 2$ and mildly discrepant for $z gtrsim 2$.
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