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Ultra-wideband (UWB) sensor networks can be efficiently applied to collect data through sensors deployed for preserving environment and rescuing lives. The high-precision positioning capabilities of UWB technology, based on its operation of ultra-short data bearing pulses, can be a reliable solution for such applications. Most of the UWB positioning techniques are based on time-of-arrival of the first path which can be effectively estimated when the probability of early detection (PoED) is accurately evaluated. In this study, we study the PoED of UWB positioning sensor networks subject to narrowband interference modelled with Nakagami-m distribution. We derive novel closed-form approximate expressions which are simpler and have remarkable accuracy compared to the previously published semi-infinite series expressions. Furthermore, to simplify our approach, we present tight lower and upper closed-form bounds for the PoED. Numerical results and simulations are performed to validate the analytical derivations.