This article presents a new spatio-temporal framework for the day-ahead probabilistic forecasting of Distribution Locational Marginal Prices (DLMPs). The approach relies on a recurrent neural network, whose architecture is enriched by introducing a deep bidirectional variant designed to capture the complex time dynamics in multi-step forecasts. In order to account for nodal price differentiation (arising from grid constraints) within a procedure that is scalable to large distribution systems, nodal DLMPs are predicted individually by a single model guided by a generic representation of the grid. This strategy offers the additional benefit to enable cold-start forecasting for new nodes with no history. Indeed, in case of topological changes, e.g., building of a new home or installation of photovoltaic panels, the forecaster intrinsically leverages the statistical information learned from neighbouring nodes to predict the new DLMP, without needing any modification of the tool. The approach is evaluated, along with several other methods, on a radial low voltage network. Outcomes highlight that relying on a compact model is a key component to boost its generalization capabilities in high-dimensionality, while indicating that the proposed tool is effective for both temporal and spatial learning.