It's long and disputed.

Firstly, it seems that the cross as a symbol predates Christianity. But this is not very surprising when you consider that it's basically two lines intersection. Beyond a circle, it's one of the simplest symbols that it's possible to create. Suggested antecedents occur in some Middle East archaeological finds, as well as the Egyptian ankh symbol.

However it's difficult to tie Christian adoption of the symbol directly to a pre-Christian precedent, because of the way early Christian grounds itself in historical claims. The crucifixion holds an important place in Christian theology from the earliest days, but the standard cross imagery does not. You can see early usage instead of the staurogram, including in manuscripts P66 (usually dated ca. 200) and P45 (dated ca. 250).

Minucius Felix, writing ca. 190-200, defends against the idea that Christians worship the cross, in Minucius Felix chapter 29. While Clement of Alexandria in Stromata, book 6, Chapter 9, refers to and interpretation of the number 318 as pointing to "the Lord's sign" clearly meaning a cross.

That said, crucifixion was a shameful death, and Christianity was not widely accepted until the 4th century, so public display and use of a cross symbol seems rather subdued until at least Constantine. I don't have good books on hand, but it seems like widespread use of the cross in Christian symbolism emerges somewhere between the 4th and 6th century, and you don't see usage of the crucifix (specifically a cross with a body portrayed on it) until the 5th century.

Tracing the stylistic developments and variations of cross imagery from that point on is a bit beyond me, but there is a firm reaction in the Reformation against crucifixes, so that there is a fairly firm division to this day that Protestants use plain crosses, not crucifixes, in their religious iconography.