This will be updated periodically as I find more to add to it.
Ruda/Shiva (Indian): I have spoken of this at length on my page Apollon and Artemis, Shiva and Paravati. Please refer there for more on this 🙂
Resheph (Canaanite): This syncretism is mentioned mainly in Cypris. If the academia that suggests Apollon came from Asia is correct (and likely the more southern part of Asia Minor given the related position of Rhodes and Apollon’s important cult center in that part of Asia Minor in Claria) then it is possible that the identification of Apollon as Resheph has some weight to it. His titles which associate him as Resheph of the garden, and Resheph of the arrow bear a lot of similarities in the character of Apollon, as do the black boundary stones which scholars who investigate this relationship attribute as primary form of Resheph and the early form of Apollon and the localities which are attributed as sacred gardens of Apollon. That both gods are associated with the healing of disease, and are connected with the deer/gazelle. Like Apollon he is also a divine charioteer, which was his most popular form in his adoption in the Egyptian pantheon.
Freyr (Norse): It seems that those who attempt to do a syncretism between Hellenic and Norse gods tend to link Apollon to Bragi, the divine musician among the gods. However in the context of Apollon in relationship to music, though he is a great musician I can’t say that this his primary function, as music and his instruments are symbols more of his cosmic nature. In Hellenismos the playing of the kithara is more linked to solar energies in which we also have Helios as a kithara player. Thus is more logical to say that the playing of the kithara was more related to his divinity as a god of light and the spiritual nature of this domain though its impact on the soul. He is the god who keeps bodies operating in harmonious movement. I find it more logical to associate Freyr with Apollon, who like Apollon is a god associated with the sun but is not literally the sun god. Like Apollon he seems to be associated with ships and the sea, weather, kingship, phallic fertility, boars, peace (harmony in other words) and the sun mostly through its fertile activity on the earth. Like Apollon he is also a warrior, though I think that this can be said of most Norse gods lol.
Heru-wer (Egypt): It is more common to see Heru-sa-Aset associated with Apollon, but this link seems made almost exclusively via the nature of healing and the killing of serpents while a babe. I consider it far more appropriate to syncretize Apollon with the elder Heru, Heru-wer, the twin brother of Set and younger brother of Wesir. He is a heavenly god whose eyes are the sun and moon respectively (which makes a good association with the lunar and solar connections of Apollon). Like Apollon he is also an inspirer of fear, a solar god, and bears a relationship to fire. Like Apollon he also bears a distinct militaristic nature. The association of Apollon with falcons (perhaps largely to do with his movement as he was considered the fastest of all the gods) can be seen in the Iliad in which the god is described as dropping from the heavens like a falcon.
Grannus (Gaul): This is a god typically associated with Apollon, so much so that one of Apollon’s epithets is believed to have an etymological connection with this name. The Germanic Gauls had consider contact with Hellas and Rome and therefore it is not surprising that this connection would be made. There may be some relationship between the Grynian Apollon mentioned in the Orphic hymn, an epithet associated with a distinct cult center in which Apollon had a temple surrounded by a beautiful garden full of fruiting and non-fruiting trees) and this Gaulic deity. Grannus is a solar god, but like Apollon is also a god of thermal springs (this is a particular well known function of Apollon in several places including Lesbos). He is likely also associated greatly with the maturation and harvest of wheat as Apollon is if we take into consideration that chants were sung to Grannus with the offering of wheat into the fire. It is speculated that Grannus gives his name to the word for grain. Grannus is likely identifiable with the Gaul Belenus who was also a solar god associated with thermal springs, and whom Apollon was considered syncretic to by Romans.