I see that this question comes up every now and again, and a couple of times recently. I really thought we had had a little discussion about it quite some time ago, but I simply can't find it with the search function or in my own long comment history.

Firstly, references to 'the cross' in Jesus first century context would almost always be understood as references to execution via crucifixion. That was the meaning of a cross for most people in most contexts, and you would need some kind of 'contextual override' to suggest it referred to something else (i.e. if you happen to translate an ancient math textbook, then yes 'cross' probably doesn't refer to crucifixion). So I don't know why you would think that time-travel Jesus wouldn't recognise it.

It's use as a symbol appears about as early as Christian iconography and symbolism does. Which is to say, not immediately; you get Christian art in general appearing more in the 3rd century onward, beginning with symbols such as fish, ship, anchors, etc., things associated with Christianity and its texts.

The usage of the cross symbol in particular was probably a unique Christian invention. Understand what that means though - two lines crossing each other is a very basic iconic depiction, so there are all sorts of 'crosses' in antiquity, but adoption of it by Christians was linked very closely with its symbolism for Jesus' manner of death, which was generally regarded as a shameful way to die in both Graeco-Roman and Jewish contexts, decding to use the cross as a symbol would not generally be regarded as 'brilliant move of marketing', except that within Christian theology there was an understanding that the very shamefulness of the death by crucifixion was part of how the Christian God took away shame. Thus, the shame-dynamic of the cross was inverted within Christian discourse.

In the 3rd century in particular Christian authors begin to refer to the cross, or the symbol/sign of the cross, as an unambiguously Christian image.

I don't really know what your question, "Was its use started by the Church, or started on its own by followers and then taken up by the Church?" is meant to mean, except that it sounds like you presuppose a degree of organisation by the early church as an Institution along modern lines are so over and against individual followers. That is really a modern construction about the relationship between the church and its members.

The Chi-Rho symbol was already in use before Constantine, but the changes associated with his rise to power and the toleration and then establishment of Christianity in the 4th century did much to cement this as a primary symbol in that century and the next, but likewise with the public dominance of Christianity in the late 4th to 5th century, Christian art and iconography began to grow and you begin to see more and more instances of the cross as a symbol.