In uncontrolled chain reactions, do atom nuclei split apart (fission) faster than they smash together (fusion), or is it the other way around?
That depends on the amount of material, temperature and pressure. My WAG would be that you can’t beat supernova fusion for high speed chain reactions.
The speed of a chain reaction is going to depend on a lot of other factors besides whether it is fission or fusion. What specifically are you fissioning or fusing? How densely are the atoms packed? etc…
Fusion isn’t really a chain reaction - each fusion reaction is only indirectly triggering other fusion reactions to occur by increasing the heat and temperature of the surrounding medium. Under some conditions such as in the core of a main sequence star fusion reactions actually have a negative effect on other fusion reactions by decreasing the density of the surrounding matter and adding non-fusable reaction products to the mixture. A supernova explosion is a special case, once you get past iron each fusion actually absorbs energy, but also increases the density and allows the gravity-driven collapse to speed up.
Fission, on the other hand, is a chain reaction - each fission reaction gives off neutrons which strike other atoms and induce them to undergo fission. Under the right conditions you actually will get an exponentially increasing chain reaction with fission, at least until the slower thermal effects catch up and blow the reacting mass apart.
Which is why such reactions are called “thermonuclear”.