We consider Bucket Elimination (BE), a popular algorithmic framework to solve Constraint Optimisation Problems (COPs). We focus on the parallelisation of the most computationally intensive operations of BE, i.e., join sum and maximisation, which are key ingredients in several close variants of the BE framework (including Belief Propagation on Junction Trees and Distributed COP techniques such as ActionGDL and DPOP). In particular, we propose CUBE, a highly-parallel GPU implementation of such operations, which adopts an efficient memory layout allowing all threads to independently locate their input and output addresses in memory, hence achieving a high computational throughput. We compare CUBE with the most recent GPU implementation of BE. Our results show that CUBE achieves significant speed-ups (up to two orders of magnitude) w.r.t. the counterpart approach, showing a dramatic decrease of the runtime w.r.t. the serial version (i.e., up to 652× faster). More important, such speed-ups increase when the complexity of the problem grows, showing that CUBE correctly exploits the additional degree of parallelism inherent in the problem.