High-bandwidth metallic coaxial nanolasers are of high interest to investigate laser physics such as thresholdless coherence transitions, and have a large variety of promising applications enabled by their ultrasmall size and large spectral bandwidth. Optical coherence properties are commonly characterized in Hanbury-Brown and Twiss experiments. However, those are difficult to perform in broadband lasers when the coherence time is an order of magnitude shorter than the temporal resolution of the single-photon detectors, thus requiring significant spectral filtering. This paper demonstrates a new approach in investigating the temporal dynamics of the photon statistics associated with the nanolaser emission, obtained without the requirement of spectral filtering. While optically pumping the nanolasers with nanosecond pulses, time-resolved second-order coherence properties are evaluated over the time duration of the pump pulse. Coherence transitions from thermal emission to lasing are observed in the gathered time-resolved photon statistics, linked to the temporal change in optical power of the nanosecond pump pulses. As nanolasers show better performance for the pulsed pumping scheme, the temporal envelope modulation of these pulses results in varying degrees of coherence within the nanolaser pulse envelope. This approach can also be readily applied to characterize a large variety of broadband lasers.