Based purely on canon information, we have no way to tell.
Following the first book, Nicholas Flamel is not mentioned again, so we have no way to tell if he died days, weeks, months, or years after the stone was destroyed. We don't know for sure if he died the day he ran out of stored elixir, or if he simply began aging again and lived out a normal lifespan afterwards.
The implications given by reading the first book are that the Elixir needs to be consumed regularly, and that if you consume it you must continue indefinitely to continue living. It is implied that, lacking Elixir, a drinker will die in short order (days or weeks, at most).
But all of this is only implied, not outright stated, and even if it were outright stated, we couldn't take it as granted.
Most of the information Harry gains about the Stone and the Elixir comes from a pair of simple references in library books (stating that Flamel created a Stone, and that it produces the Elixir of Life) and what he is told by Hermione and Dumbledore.
Hermione, a Muggleborn, only knows what she could learn from published books. She has no sources of rare or exclusive knowledge, as she is limited to what is publicly available in the Hogwarts Library (not even the Restricted Section) and booksellers like Flourish & Blotts. Obviously, public information about the Stone and its capabilities (and limitations) would be possible to be misleading or flat-out wrong: Flamel and his wife have a vested interest in keeping their Stone's capabilities secret and the weaknesses (if any) of their immortality unknown.
Dumbledore, as an alchemist of note and a personal friend of Flamel's, obviously knows more. That said, Flamel is famously against sharing information about the Stone. Dumbledore has been repeatedly shown to be willing to keep secrets for others, especially when they are important secrets. He is also known to lie to Harry on occasion or to withhold critical information. This is especially true in the first book.
Thus, not only do we NOT know for sure how the Elixir works, we cannot even be certain that the information we DO have about the Stone is accurate in anything but the broadest of strokes.