Robust scheduling for an integrated electricity and heat system (IEHS) is essential to hedge the uncertainties brought by volatile wind power and heat load that could pose critical security threats to IEHS operation. Because of the distributed structure of different energy sectors, their models are managed locally and private information must be protected. In this context, a decoupled solution is favorable for robust IEHS scheduling. However, due to the specific problem structure with a nonseparable Lagrangian, conventional iterative distributed algorithms are not applicable. In this paper, we make the first attempt for distributed robust IEHS scheduling by providing a non-iterative decoupled solution. A two-stage robust model considering electricity and heat uncertainties is formulated, and the inapplicability of conventional algorithms in decoupling this model is analyzed in depth. To facilitate feasible region projection, a network reduction method considering heat uncerntainties is then devised. Finally, an efficient non-iterative decoupled solution is proposed with guaranteed optimality while preserving the operational independence of different energy sectors. Case studies in two test systems validate the effectiveness of the proposed method.