In the future, the nuclear threat is less likely from a massive nuclear attack, but more likely from crude devices, such as dirty bombs perpetrated by an individual or group of terrorists. Detection and prevention of these nuclear devices are critical to the safety and security of the general population. In general, the research of detecting various radioactive sources using individual sensors has been well established in terms of both detection devices and detection methods, most of which are dedicated to single or co-located sensor systems. Large monitoring systems at choke points (e.g. commercial airports and harbors) can prevent the entry or exit of nuclear sources. However, it cannot protect a perimeter that spans a large area, such as land and sea borders. Furthermore, in many practical scenarios, it is desirable to detect low-level radiation and identify low-level radioactive sources. Recent advances in sensor network technologies have opened up the potential for improved detection, as well as the estimation of source parameters, by utilizing measurements from multiple, geographically dispersed sensors. Different from the existing works on radiation detection using sensor networks, in this paper, we focus on the quickest detection method that would identify nuclear radiation as soon as possible after it occurs, while keep the false alarm rate low. Specifically, we propose that each sensor performs a nonparametric version of the Page's Cumulative Sum (CUSUM) test based on its local measurements, since the occurrence of the nuclear radiation is unpredictable and we assume that we do not have prior knowledge of the adversary (e.g., who carries a dirty bomb). Then the local decisions from multiple sensors are sent to a fusion center for combining and a final decision is made. We present numerical results to demonstrate the effectiveness of the proposed scheme using the experimental measurements from a COTS detectors in our lab.