Uncertainty quantification for PDEs involves numerous queries, demanding a significant number of model evaluations and substantial computational resources. In response, many surrogate and reduced-order models have emerged, often incorporating multi-fidelity approaches to address this challenge. With the rise of deep learning surrogates, it becomes crucial to manage computational resources effectively, starting from the sampling of high-quality simulations to the actual training of the model. Here, we shall present a novel strategy for the optimal management of computational resources, termed ”Deep learning enhanced multi-fidelity Monte Carlo” (DL-MFMC), providing domain practitioners with precise guidelines on how to split the computational burden. To demonstrate its efficacy, we apply DL-MFMC to a 3D-1D multiscale computational model of microcirculation, expediting the estimation of biophysical metrics related to oxygen transfer and radiotherapy outcomes. Experimentally, our approach showcases significant speed-ups and a substantial reduction in overall computational costs compared to traditional Monte Carlo.