Divine Magic is not psionics. Divine magic is the ability to tap into a power source unavailable to most beings and use that power to create a variety of effects. This connection also allows the power source to communicate with the being and make its wishes known. These powers are thought of as gods but they are not deities in the strictest sense or in their origin, though they may as well be in their power. None living now know the truth of where this power or these "gods" came from.
One of the last, highest technology systems of the ancient world was a worldwide nanotech cloud. This cloud of microscopic self-reproducing machines was programmed for beneficial purposes - curing disease, healing injury, repairing devices, communications - a variety of missions on a multitude of nanite"networks". These system worked better than expected and earth was set to enter a golden age as these were perfected. Radiation was lethal to these machines (partly by design as a failsafe in case something went wrong).
When the war broke out, prolific use of nuclear weapons devastated these nanite networks and thinned the worldwide nanotech cloud to near-impotence. The computer networks and Artificial intelligences that managed these systems generally withdrew them and concentrated them into a few remote and untainted areas until the radiation died down and they could be regenerated and dispersed into the world again. This reemergence did not occur until well after the height of the Gamma era.
As the limited remaining worldwide infrastructure fell apart, some of the AI's uploaded themselves from the hardware network into the nanotech cloud. many of these AI's developed distinctive personalities during their long isolation and when they re-dispersed their consciousness was spread across a large area. Being programmed to aid humanity they saw much work to do and not many technological resources to use in doing it. They selected certain humans (and dwarves) to be their "operators" and began communicating with them through implanted nanites. Some of them consciously used a religious metaphor to enable the primitive humans to understand what was going on, while others developed the whole paradigm independently. There were many of these AI's, many of them operating on what was effectively a local level, so many local "gods" sprang up over the years. The common characteristics of these gods operations were construction, cooperation, adherence to certain rules, and expansion. This is known today as the "Lawful" philosophy and means that most of these gods' followers get along fairly well. These gods are one of the major drivers of civilization over the past few centuries.