Where did the nick of time come from?

Answer 1

Around the 1580's the phrase in the nick meant in the cut or notch to mean at a precise moment in time. The phrase evolved to be in the nick of time. Nowadays the meaning has also evolved to mean at the last moment.
At the last second. Example: I was begging to starve but in the nick of time, the Pizza man delivered our food.

Answer 2

Sometime round about the 1580s the phrase in the nick or in the very nick began to be used for the critical moment, the exact instant at which something has to take place. The idea seems to have been that a nick was a narrow and precise marker, so that if something was in the nick it was precisely where it should be.

It seems that users of the expression pretty soon afterwards found this association of ideas needed some elaboration, so started to add of time to the expression, and that's the way it has stayed ever since. These days, the phrase more usually refers to something that only just happens in time, at the last possible moment.