We numerically investigate the energy and arrival-time noise of ultrashort laser pulses produced via resonant dispersive wave (RDW) emission in gas-filled hollow-core waveguides under the influence of pump-laser instability. We find that for low pump energy, fluctuations in the pump energy are strongly amplified. However, when the generation process is saturated, the energy of the RDW can be significantly less noisy than that of the pump pulse. This holds for a variety of generation conditions and while still producing few-femtosecond pulses. We further find that the arrival-time jitter of the generated pulse remains well below one femtosecond even for a conservative estimate of the pump pulse energy noise, and that photoionisation and plasma dynamics can lead to exceptional stability for some generation conditions. By applying our analysis to a scaled-down system, we demonstrate that our results hold for frequency conversion schemes based on both small-core microstructured fibre and large-core hollow capillary fibre.