If Borrowed Time leaves the battlefield before its enters-the-battlefield ability resolves, the target permanent won't be exiled.

Auras attached to the exiled permanent will be put into their owners' graveyards. Equipment attached to an exiled creature will become unattached and remain on the battlefield. Any counters on the exiled permanent will cease to exist.

If a token is exiled, it ceases to exist. It won't be returned to the battlefield.

Borrowed Time's ability is a single ability that creates two one-shot effects: one that exiles the permanent when the ability resolves, and another that returns the exiled card to the battlefield immediately after Borrowed Time leaves the battlefield.

In a multiplayer game, if Borrowed Time's owner leaves the game, the exiled card will return to the battlefield. Because the one-shot effect that returns the card isn't an ability that goes on the stack, it won't cease to exist along with the leaving player's spells and abilities on the stack.