We investigated the timing jitter of superconducting nanowire avalanche photodetectors (SNAPs, also referred to as cascade switching superconducting single photon detectors) based on 30-nm-wide nanowires. At bias currents (IB) near the switching current, SNAPs showed sub 35 ps FWHM Gaussian jitter similar to standard 100 nm wide superconducting nanowire single-photon detectors. At lower values of IB, the instrument response function (IRF) of the detectors became wider, more asymmetric, and shifted to longer time delays. We could reproduce the experimentally observed IRF time-shift in simulations based on an electrothermal model, and explain the effect with a simple physical picture.