The combined operations of power distribution network (PDN) and district heating network (DHN) can enhance the flexibility and improve the overall energy efficiency of power systems. This article implements a rolling look-ahead unit commitment scheme in a combined PDN and DHN to exploit the operational flexibility of rapid-response combined heat and power (CHP) units under significantly variable renewable energy source (RES) power output. The scheme is formulated as a multistage distributionally robust (DR) unit commitment model that respects the non-anticipativity of decision variables for sequential revelations of uncertainties. In contrast to the moment-based ambiguity sets employed in conventional DR models, the proposed framework constructs an ambiguity set based on probabilistic forecasts. In this regard, a compatibility is achieved between DR approaches and probabilistic forecasts by incorporating comprehensive distribution information of RES power output stemming from probabilistic forecasts into DR models. The computational challenge associated with the proposed multistage DR model is addressed by applying linear decision rules. Moreover, a new constraint reformulation approach is utilized to increase the computational tractability. The proposed model will ultimately cast into a tractable mixed-integer linear programming problem. The effectiveness of the proposed method in capturing a comprehensive distribution of RES power output and reducing the the combined system operation cost is demonstrated by case studies carried out on the Barry Island multi-carrier energy system. Numerical results also validate the proposed model's improved computational performance.