So, you want to create a god. Your own god. A god omniscient and omnipresent within a manageable sphere of influence that can talk to other gods. I thought about that a lot, and found ways to achieve telepathy. Here's one of the ways to do it.
A telepathic alien species from the Sci Fi shortfilm under the same name by Kaleb Lechowski.
A simple cephalic anatomy of the creature R'ha, showing its communication organs.
Basically, they have a Wi'Fi embedded in their heads, with a receiver and a transmitter fitted as biological organs near the brain. And you know how Wi'Fi works like how information is translated from 1's and zero's into electronic signals then into changes in the ups and downs in the intensity of the carefully emitted radiowaves. The reverse process works to decode the transmitted information.
Though the specifics of R'ha telecommunication is left out, it's safe to assume they not only detect changes in the intensity of radio waves, but changes in frequencies. This means they hear radiowaves just as how we see colors.
Based entirely on speculation, their head is shaped to filter EM waves and the filtered waves enter their resonance sensor to detect long-wave transmissions so that the brain can interprets the structure of the long-wave. This interpretation of detection can be a language of its own, mapping letters, words, phrases, sentences, and who knows maybe even whole mental snapshots containing memories and ideas and feelings, into changes in intensity, frequency, modulation, phase shifts, or other signal processing techniques.
The emitter on the other hand, contains a different resonator of its own, translating synaptic firing patterns and neural signals directly into the same changes in radio waves. Its just like the vocal chords, but with even tinier muscles and some metallic components that allow vibration in a wider range of frequencies.
With organs to detect, interpret, and create radio waves, R'ha are practically Wi'Fi-enabled, and telepathy is justified.
But you might be thinking. *HAHA. They're so ugly. So...
No. Your gods won't look like that.
We can have similar organs but put them somewhere else. Better yet, we can reconfigure parts of our existing organs to be just as sensitive and controllable as the R'ha's.
More like... this
Sorry, no picture yet. But it works like this.
1. Pentachrome Eyes
You already know about people who have four cone cells in their eyes, who can see 1000 times more color than usual, usually towards the bluer side of things. But imagine having five, with one for far ultraviolet (bordering on XRay) and another for far infrared (bordering on radio waves)? Not just incredible detail, but the temperature of things is visible to you as well. So there, you have it. Ultra-awareness.
For the telepathy, this is just the ultra-awareness, but coupled with ultra-interpretative learning. Since you can see radio waves and x-rays, you can even peer into the internal structure of objects with just your eyes, kinda like superman on both heat and x-ray vision mode. I wonder what can happen if you can see entangled photons as well.
2. Hyper-octave Laryngeal Muscles
Just like the R'ha's emitter, you have vocal chords, but ones mutated solely for a wider range of frequencies just as your pentachrome eyes can see in that range. Maybe a number of superfast-twitch muscles akin to humming bird wing muscles, but tailored for making membranophone sounds. Together with a modified uvula that works like a transmitter antenna that can vibrate, generate, and detect electromagnetic wave, you can now radiate the same imaginary sounds and colors that you can see with your pentacolor-sensing eyes.
Electromagnetic Ultra-awareness + Bio-Wi'fi = Telepathic Radar
With these two, you're not only wi'fi-enabled, but your awareness spreads like a radio broadcast originating from the roof and the root of your tongue, like a spoken word shouted as an EM wave corresponding with your name. Everything that your word/name/shout/broadcast touches, sends an echo to your eyes and ears, but slightly changed in a way that tells you something about whats material and at what distances the echoes came from, like diffractograms, radar, sonar, and the likes. From then on, your mind can be trained to build a composite image of your surroundings, with the quality and resolution fading after a few meters depending on how loud you shouted/how strong the broadcast is/how experienced you are at this type of radio-echolocation.
Similarly, someone else can speak something with those modified uvula and vocal chords such that only other radio-communicating creatures and devices will hear it, but only another person that knows the correct frequency/channel can understand how to subconsciously decode it.