In real history, we have two examples of how something similar was done, so that we cannot say how to do it in general, only how it worked in the Antiquity.
In the Antiquity, in general
In the Antiquity, in Europe and in North Africa, and in the Near and Middle East, just about everybody was a polytheist, with the one well-known exception, and it was generally and commonly accepted that different peoples had different gods. A Greek would not have expected an Egyptian to worship the Greek gods; an Egyptian would not have expected a Babylonian to worship Egyptian gods; a Babylonian would not have expected a Hittite to worship Babylonian gods. Moreover, it was understood that religion was something a person imbibed with their mother's milk, and was not expected to change; so that foreigners who settled in another city were understood and expected to continue worshipping their original gods.
All this worked well as long as peoples didn't mix too much, that is, as long as foreigners remained foreigners, and as long as nobody went about claiming that their God or gods were the only (or even the topmost) God or gods. Who cares that the foreigners worship? They are foreigners.
In the Antiquity, specifically: interpretatio Romana
But then came the power of Rome, and, for its own imperial purposes, Rome introduced the idea that citizenship and allegiance are distinct from ethnicity; for the first time in history, they accepted that one did not have to be born a Roman, one could become a Roman; and, even if one didn't want to become a Roman, they owed allegiance to Rome. A side-effect of this admirable openness was that it became somewhat important to put a little order in this business of multiple concurrent sets of gods.
Note that Rome did not reach such a progressive state right from the beginning. Initially, the Romans were as wary of foreign gods as anybody else; one of the oldest monuments of the Latin language is the famous Senatus consultum de Bacchanalibus, the Decree Against Bacchanals of 186 BCE, showing the early Roman state in full repressive mood against the spread of a foreign, in this case Greek, cult.
But, as the territory under Roman Power (that's what Imperium Romanum actually means) expanded, this repressive position became obviously counterproductive, so the Romans appropriated a Greek idea and elevated it to the rank of political principle. The Romans did that quite often; Graecia capta ferum victorem cepit, as one of their poets said: captive Greece captivated her rude vanquisher.
The idea in question was the interpretatio Graeca, a device used by Greek historians to explain foreign religions to a Greek audience. In its original Greek instance, this consisted in drawing parallels between a Greek deity and a foreign one, so that, for example, Herodotus could explain to his readers that the Egyptian gods Amon, Osiris or Ptah were sort-of like Egyptian variants of Zeus, Dionysus and Hephaestus. They weren't the same, there were differences, but they could by understood by comparison.
The Romans made on more step, and in their officious (yet never "official") interpretatio Romana, they made foreign gods notionally the same as Roman gods. Odin was sort-of like Mercury, because he was smart and cunning and had a winged horse; sort-of like? No, he was Mercury, it's just that those barbarian Germans used a different name. Thor was sort-of like Jupiter, because he wielded thunderbolts; why not make him to actually be Jupiter? And similarly, Tíw (Týr, Ziu) was only a name of Mars, and Freya a name of Venus; which we can still see in the names of the week, which in Germanic languages go Tues-day, Wednes-day, Thurs-day, Fri-day, with the Germanic names of the gods, paralleling the Romance Marte-dì, Mercole-dì, Giove-dì, Vener-dì, which continue the Roman names of the gods.
Quot hominum linguae, tot nomina deorum, wrote Cicero in his book about the Nature of the gods: the gods have as many names as humans have languages.
How did they reach this state? It all began with the Romans' exposure to the Greek mythology.
The Roman and the Greek religions were very different, with remarkably few points of similarity, other than the general look (they were both luxuriantly polytheistic) and a very small number of shared deities of Indo-European inheritance. Jupiter and Zeus are really the same, as are Vesta and Hestia, or Aurora and Eos. But Minerva is profoundly different from Athena, Venus is definitely not Aphrodite, and Mars shares with Ares only their combative attitude.
Nevertheless, the Greek religion had something the Roman religion didn't, namely, mythology. The Roman native deities are more in the nature of abstract ideas, they don't look like humans, they don't behave like humans and they most definitely don't have thrilling adventures in the human world. Boring. So, for literary and artistic purposes, the Romans imported the Greek mythology wholesale, and used it as if it was their own, making the obvious divine identifications. In the state religion, in the stately rituals, the original conception prevailed and remained active, but in poems and novels, in the statuary, and in decorative pictures, the Roman names were applied to the much more adventurous and relatable Greek gods.
To exemplify, around the middle of the 3rd century BCE, Livius Andronicus renders the first line of the Odyssey (ἄνδρα μοι ἔννεπε, Μοῦσα, πολύτροπον, andra moi ennepe, Mousa, polytropon, "tell me, Muse, of the well-versed man") as virum mihi Camena insece versutum, seeking to establish an equivalence between the Roman Camenae and the Greek Muses; but one hundred years later, Ennius writes shamelessly in his Annals: Musae quae pedibus magnum pulsatis Olympum, "Muses, who with your feet beat mighty Olympus"; thus giving up any pretense that a Roman audience did not know who the Muses were or what the Olympus mountain was.
Once the Greek gods were accepted as equivalents of the Roman gods, although everybody knew they weren't, it was not hard to extend this to the other peoples of the Empire. Where the Greek went, the Gauls, and the Germans, etc. followed.
In later times, the Romans started accepting that there were deities which didn't even have Roman names; and in Rome the cult of Cybele and Isis prospered. Once a dividing wall is breached, there is no division any more.
It should be noted that this identification of foreign gods with Roman gods was purely notional; it had exactly zero effect on the ground in those lands were those foreign gods were actually worshipped. The Romans did not go to Cyprus to explain to the natives that their Aphrodite had been officially identified with Venus and the cult of Aphrodite in Cyprus had to conform to the cult of Venus in Rome. They did not go to Germany and explain that Woden / Odin had been identified with Mercury and the Germans should immediately make him a god of thieves and merchants. No; the interpretatio Romana was for the use of Romans, so that they would feel at home in a diverse world.
As long as the peoples of the Empire agreed to build a temple for the Emperor and to perform the required rituals, all was fine. Nobody went into silly theological disputations about which god came first, or which god is paramount; such discussions were considered light entertainment, suitable for a pleasant afternoon with friends and a crater of mulled wine.
