Perdido Beach nuclear power comes from a nuclear power plant located at the exact center of the FAYZ. It supplied Perdido Beach with electricity. Before the FAYZ, one of the plants reactors was hit with a meteor (the meteor was most likely the Gaiaphage). The meteor sealed the radioactive material deep underground, so the owners of the plant paved over it, believing it was safe. According to Astrid, this radiation is what caused the mutations in the FAYZ.
Even after the FAYZ, the plant kept running until it was shut down by Computer Jack during the Battle for the Nuclear Power Plant, and the uranium was taken by Caine while under the Gaiaphage's influence. The destruction of the power plant (the plant itself was severely damaged during the battle, so it counts as destroyed as it wont be possible to fix the damages) marks the beginning of the Dark Time, where as there's no electricity powered sources in Perdido Beach.