False vacuum decay (FVD) plays a vital role in many models of the early Universe, with important implications for inflation, the multiverse, and gravitational waves. However, we still lack a satisfying theoretical understanding of this process, with existing approaches working only in imaginary (Euclidean) time, and relying on numerous assumptions that have yet to be empirically tested. An exciting route forward is to use laboratory experiments which undergo transitions analogous to FVD, allowing nature to simulate all of the non-perturbative quantum effects for us. In this talk, I will discuss ongoing work to develop such analogue FVD experiments within the Quantum Simulators for Fundamental Physics (QSimFP) consortium. In particular, I will present numerical lattice simulations of ultracold-atom systems undergoing FVD, and discuss outstanding challenges in using upcoming experimental data to understand the early Universe.