In this paper, we consider distributed utility-maximizing rate allocation in cyber-physical multihop wireless networks carrying prioritized elastic flows with different end-toend delay requirements. This scenario arises in military wireless networks (dominated by audio and video flows) that must satisfy end-to-end deadlines. Due to the inherent difficulty in providing hard guarantees in such wireless environments, the problem is cast as one of utility maximization, where utility depends on meeting deadlines. Based on a recent result in real-time scheduling, we relate end-to-end delay of prioritized flows to flow rates and priorities, then impose end-to-end delay constraints that can be expressed in a decentralized manner in terms of flow information available locally at each node. The problem of utility maximization in the presence of these constraints is formulated, where utility depends on the ability to meet deadlines. The solution to the network utility maximization (NUM) problem yields a distributed rate control algorithm that nodes can independently execute to collectively maximize global network utility, taking into account delay constraints. Results from simulations demonstrate that a low deadline miss ratio is achieved for real-time packets, without significantly impacting throughput, resulting in a higher total utility compared to a previous state-of-the-art approach.