World Quests were introduced with the Legion expansion, and each expansion since has added new World Quests to the game. For each expansion world quest access is unlocked in a different way.
Dragonflight world quests require that you complete the Main Storylines for all zones and have the achievement "Just Don't Ask Me to Spell It". https://www.wowhead.com/achievement=16363
Once you unlock Dragonflight world quests with one character on your Battle.net account, alternate characters can access them as soon as they arrive in a Dragonflight zone.
Shadowlands World Quests require that you reach level 60 with one character and you join a Covenant.
Once you unlock Shadowlands World Quests with one character on your Battle.net account, alternate characters can access them as soon as they join a Covenant, which will be possible at level 50 (down from 60). World Quests in Korthia are an exception and still require the alternate character to reach level 60 and unlock Korthia.
Battle for Azeroth World Quests require that you reach level 60, you obtain your Heart of Azeroth, and you complete the quest Uniting Kul Tiras (Alliance) or Uniting Zandalar (Horde) which requires that you reach Friendly status with the factions of Kul Tiras or Zandalar.
Once you unlock Battle for Azeroth World Quests with one character on your Battle.net account, alternate characters can complete Uniting Kul Tiras / Zandalar ignoring the reputation requirement. Alternate characters are not required to obtain the Hearth of Azeroth; however, World Quests that reward Azerite will reward nothing to characters that don't have the Heart of Azeroth.
Legion World Quests require that you reach level 45 and complete the quest Uniting the Isles. This quest requires that you reach Friendly status with all the factions of the Broken Isles.
Once you unlock Legion World Quests with one character on your Battle.net account, alternate characters can complete Uniting the Isles ignoring the reputation requirement.