If an Earth-sized or larger planet had sufficiently large amounts of radium, thorium or uranium, couldn't the resulting fission trigger periodic fusion of hydrogen? I'm thinking of fission happening at great depths and high pressure, with some hydrogen-containing material nearby to the radioactive elements.
First of all, there will certainly occur (very) few fusion reactions, as you always have some particles in the tail of a Maxwellian that have the correct energy.
Speaking, however, of a burning plasma, like our sun, that is not going to happen. To achieve a burning plasma (self-sustained in that sense that all energy required to achieve fusion is delivered by the fusion reaction itself), you need to basically fulfill the Lawson criterion: you need enough particles (density) with enough energy (temperature) and that for a sufficient time (confinement time).
The sun is doing that mostly with a very high density (due to the gravity). On earth, the most promising fusion approach seems to be to achieve high temperatures instead (magnetic confinement fusion).
As you asked about fusion in a planet's core, let's use the sun's analogy: a huge mass is required for the gravitational pressure needed to have a burning (in the sense of self-sustained fusion) core. What is the heaviest planet in the solar system? That is Jupiter. Still, the mass and the resulting gravitational pressure is too low. As is turns out, though, if Jupiter would be heavier by approximately 1 order of magnitude, see here, its core would "ignite" (and it would be called a brown dwarf).
So, if you asked about the occasional fusion reaction (where a fission reaction might be helpful for some initial heating), that might happen; the transition into a burning plasma will certainly not happen.