Multiphoton interference is at the heart of photonic quantum technologies. Arrays of integrated cavities can support bright sources of single photons with high purity and small footprint, but the inevitable spectral distinguishability between photons generated from nonidentical cavities is an obstacle to scaling. In principle, this problem can be alleviated by measuring photons with high timing resolution, which erases spectral information through the time-energy uncertainty relation. Here, we experimentally demonstrate that detection can be implemented with a temporal resolution sufficient to interfere photons detuned on the scales necessary for cavity-based integrated photon sources. By increasing the effective timing resolution of the system from 200 to 20 ps, we observe a 20% increase in the visibility of quantum interference between independent photons from integrated microring resonator sources that are detuned by 6.8 GHz. We go on to show how time-resolved detection of nonideal photons can be used to improve the fidelity of an entangling operation and to mitigate the reduction of computational complexity in boson sampling experiments. These results pave the way for photonic quantum information processing with many photon sources without the need for active alignment.