When Christians began using a cross as an identifier is a separate issue in my mind. Though your suggestion that it was inherited as the Tav is interesting. I'm foremost interested in the formation of the doctrine of crucifixion. What were the influences and rational for the belief that the Christ was destined to be hung on a tree/xylon. Traditionally it's assumed a crucifixion occurred and that was later perceived to fulfill "prophecy". However, the opposite is more consistent with the process at work in the Gospels. That is, the story was written using elements drashed from the Tanakh. The tree/wood from Deut. and Genesis seems reasonably to be the inspiration for understanding the Christ/Son figure being killed on a tree, likely (and as proposed by Doherty Et Al.) this was in a lower level of heaven or briefly and imperceptively on earth.
I'm of course also admitting the influence from myths of other deities like Attis and Silenus who were hung on a tree contributed to this anticipation/revelation. Again, perhaps a short time later this was reinterpreted as a Roman execution by crucifixion on a cross per a number of NT writers. And, as you suggest perhaps the familiarity with the Tav symbolism made it all irresistible.