Nuclear reactions happening in the Earth are all radioactive decays. These produce heat, and it's in fact the main contribution (I've written another comment here with a bit more detail) but most of it happens in the continental Crust and in the Mantle.

This nuclear process is not generally dependent on pressure, in fact it mainly happens in the external layers of our planet. The distribution of radiogenic heat depends on where the element that may undergo beta-decay are located. If you are curious, the main elements responsible for this on Earth are Uranium, Thorium and Potassium.

Despite there being large uncertainty, geochemical studies show that these element are not much soluable in the liquid core. Radioactive decay contributes to the heat production in the core for 0.2 TW, while the the total amount of heat produced by the core is around 10-15 TW.

EDIT: well, not all nuclear reactions in the earth are decays. A few occurrences of natural fission reactions have been found, where particular conditions, like more abundance of a certain uranium isotope and different amount of oxygen in the atmosphere, made the fission reactions possible. The only few occurrences known happened in the far geological past in Oklo (region of Gabon).

Not caused by the pressure (that's what stars do and the Earth isn't one) but rather the heat of radioactive decay. It's also believed that tidal forces from the Moon create some heat.