Clerics in OD&D are implicitly Christian (their imagery is based on medieval Christian crusader-priests, they have level titles like Acolyte, Vicar, Curate, and Bishop (and Lama, which always seemed odd...), they carry crosses for holy symbols, and most of their spells are obviously derived from Old and New Testament miracles). Even Supplement IV (Gods, Demigods & Heroes) doesn't really change this -- there's no provision in that book for clerics actually worshipping any of the included deities.
In Basic D&D (at least up to the release of the Gazeteer series) clerical religion is handled abstractly -- IIRC the Mentzer Basic Set goes so far as to specifically advise the DM not to assign specific religions and patron deities (or at least not real world-derived ones) to cleric characters.
In AD&D we'd usually assign our characters patron deities (even the non-clerics -- there's a spot for it on the old goldenrod AD&D character sheets after all) chosen from D&Dg or the World of Greyhawk set but it was only window-dressing -- we didn't, for instance, grant clerics different spells or abilities based on what deity they worshipped (not even the suggested stuff from the WoG set). There also wasn't any attempt at consistency -- deities were assigned based on "coolness" and alignment, rather than by culture, region, etc.
It wasn't until the 2E era that I started trying to craft an actual pantheon for my game-world.