Did Alexander impose his Greek polytheism on the Persians he conquered or did the Hellenization happen naturally over the course of time and Greek soldiers and traders living among them? Or was it something else?
It was a combination of both.
Alexander the Great encouraged the spread of Greek culture as noted by Plutarch in his work On the Fortunes of Alexander:
But if you examine the results of Alexander's instruction, you will see that he educated the Hyrcanians to respect the marriage bond, and taught the Arachosians to till the soil, and persuaded the Sogdians to support their parents, not to kill them, and the Persians to revere their mothers and not to take them in wedlock. O wondrous power of Philosophic Instruction, that brought the Indians to worship Greek gods, and the Scythians to bury their dead, not to devour them! DWe admire Carneades' power, which made Cleitomachus, formerly called Hasdrubal, and a Carthaginian by birth, adopt Greek ways. We admire the character of Zeno, which persuaded Diogenes the Babylonian to be a philosopher. But when Alexander was civilizing Asia, Homer was commonly read, and the children of the Persians, of the Susianians, and of the Gedrosians learned to chant the tragedies of Sophocles and Euripides. And although Socrates, when tried on the charge of introducing foreign deities, lost his cause to the informers who infested Athens, yet through Alexander Bactria and the Caucasus learned to revere the gods of the Greeks
While Alexander is known for kick-starting Hellenization in Asia, he did not live long enough to have a significant effect. It was in fact, the people who succeeded him (Greek migrants, Intellectuals, successor kings) who were the ones that continued the process and gave a lasting effect.
Migrants who came from the Greek states and settled in Asian settlements brought with them their culture and religion. Natives often assimilated into this Greek culture and often opposed it.
The Greeks who came to Asia did not however stamp out the religions and cultures of the natives. In some cases they actually adopted foreign elements into their own religion. There are several examples of Greek deities who have mixed Greek/barbarian origins such as Serapis.
Alexander the Great did NOT proselytize, nor impose the Olympian religion on his subjugated lands, including the Persian people.
The centuries old Persian Zoroastrian religion continued 1000 years into history until the arrival of Islam-(via the Arabs). It was essentially Islam which replaced Persian Zoroastrianism and only a tiny percentage of indigenous Iranians are still adhering to the ancient faith. There are also the "Parsis"-(or Persian Zoroastrians) of India, who resettled into the Indian subcontinent after the arrival of the Arab Muslims into Persia.
Alexander's pro-Hellenism campaign did not seek to convert "barbarians"-(or Non-Greeks) to the Olympian religion. Although Alexander believed that having a knowledge or familiarity with the Olympian religion was important for the lands and peoples he conquered, religion, was not at the forefront of his campaign and mission. As far as I know, none of the lands and peoples Alexander conquered were forcibly converted to Olympianism-(Alexander The Great was unlike the Spanish Conquistadors or Turkish Sultans, who did convert many conquered lands and peoples to their respective religions).
For Alexander, it was the Greek language that was paramount whereby the "barbarians" needed to learn Greek-(both as a formal and informal language); and in a way, Alexander's Greek linguistic legacy did permeate many parts of his empire for several centuries.