In all honesty, I think it actually might have been neither who fired first, but a more global conspiracy, made by none other than Vault-tec. Since they had all this power, making all these underground bases to do experiments on humans, it would only make sense that they would WANT a nuclear war so they can start their mission.
Think about it: The G.E.C.K. super mutants, attempted human control, cloning, inbreeding. All different experiments. Yet at the main Vault tec vault, it could all be monitored, so that they could find the info they needed for literally hundreds of years. Sure it meant billions would die, but hey, when did a greedy, power hungry corporation care about any collateral damage.
Think on that.