Ultracold dilute gases represent one of the most powerful platforms for quantum simulation, owing to the unparalleled accuracy and precision with which they can be manipulated via electromagnetic fields. A prominent application is found in optical lattice clocks, where such control enables state-of-the-art metrological performance. The theoretical description of the out-of-equilibrium dynamics of these systems poses a significant computational challenge, which has been recently addressed by the open time-dependent variational Monte Carlo (otvmc) method [arXiv:2506.23928], capable of providing numerically accurate simulations of open many-body quantum dynamics. In this thesis, we extend the otvmc framework to encompass the dynamics of open quantum systems subject to continuous measurement, substantially broadening the class of experimentally relevant regimes accessible to numerical investigation.