A multilevel Monte Carlo ensemble (MLMCE) coupled with the finite element method is applied to address numerically transient heat equations characterized by random diffusion and Robin coefficients. By incorporating two ensemble averages for the Robin boundary and diffusion coefficients, we present an extended Monte Carlo ensemble scheme tailored for the uncertain transient heat equation. The suggested MLMCE approach resolves a single linear system that entails multiple right-hand-side vectors for a group during each time step, thereby decreasing both the storage requirements and the computational expenses associated with the solution process. Stability analysis and error estimates of the method are derived under some conditions involving two ratios between fluctuations in the thermal conductivity and a random Robin coefficient corresponding to their mean. Numerical experiments are presented to confirm the theoretical results and verify the feasibility and effectiveness of the proposed approach.