We propose two pipelines for convex optimisation problems with uncertain parameters that aim to improve decision robustness by addressing the sensitivity of optimisation to parameter estimation. This is achieved by integrating uncertainty quantification (UQ) methods for supervised learning into the ambiguity sets for distributionally robust optimisation (DRO). The pipelines leverage learning to produce contextual/conditional ambiguity sets from side-information. The two pipelines correspond to different UQ approaches: i) explicitly predicting the conditional covariance matrix using deep ensembles (DEs) and Gaussian processes (GPs), and ii) sampling using Monte Carlo dropout, DEs, and GPs. We use i) to construct an ambiguity set by defining an uncertainty around the estimated moments to achieve robustness with respect to the prediction model. UQ ii) is used as an empirical reference distribution of a Wasserstein ball to enhance out of sample performance. DRO problems constrained with either ambiguity set are tractable for a range of convex optimisation problems. We propose data-driven ways of setting DRO robustness parameters motivated by either coverage or out of sample performance. These parameters provide a useful yardstick in comparing the quality of UQ between prediction models. The pipelines are computationally evaluated and compared with deterministic and unconditional approaches on simulated and real-world portfolio optimisation problems.