Zakat's obligation isn't actually tied to a person's age at all — it hinges on wealth reaching the nisab (minimum threshold) and being held for a full lunar year (hawl), not on how old the owner is. As reported from the Prophet ﷺ, "There is no zakat due on wealth until a year has passed over it" (Sunan Ibn Majah 1792, narrated by Aisha رضي الله عنها — its individual chain is debated by some hadith scholars, but the hawl requirement itself is agreed upon by scholarly consensus). So for an adult, once they are Muslim, free, and their wealth reaches nisab — conventionally 595g of silver or 85g of gold, or the cash equivalent — and stays there for a full lunar year, zakat becomes fard on them, whether they are 18 or 80.
Where age genuinely enters the picture is a real, live disagreement between the schools over whether a minor even needs to be an adult for zakat to apply to their wealth at all:
The Hanafi school holds that puberty (bulugh) is itself a condition for zakat on wealth — so a minor's savings, however large, are not zakatable until they reach maturity, the same way a minor isn't yet obligated to pray or fast.
The majority — the Maliki, Shafi'i, and Hanbali schools — hold that maturity is not a condition for zakat on wealth specifically. If a child's (or a person of unsound mind's) wealth reaches nisab and a year passes over it, zakat is due every year, paid out of that wealth by their guardian (wali) on their behalf. Their evidence includes a report from Umar ibn al-Khattab (رضي الله عنه), preserved in Imam Malik's Muwatta (Book of Zakat, 17:12): "Trade with the property of orphans, so that it isn't consumed by zakat" — a saying of Umar, not a hadith of the Prophet ﷺ, but taken by these schools as evidence that a minor's or orphan's wealth was already understood to be zakatable in the earliest generation.
So the honest short answer is: there's no birthday that flips a switch. What matters is the wealth itself crossing nisab and being held for a year — whether that can happen before puberty depends on which school's fiqh a person follows. Use the Zakat Calculator to work out what's owed on your own wealth, and see the Zakat section on Practice for the fuller explanation, including agricultural and livestock zakat.