Would the Romans (or the Jewish leadership) having just executed the leader of what they thought was a dangerous cult, have given followers/supporters of that leader the access to the instrument of His death, let alone the opportunity to take pieces of it with them?
Iirc, at the end of the Civil War, Conferate regiments were ordered under the term of surrender to give up their flags/standards. Lest they be retained (or displayed) for symbolically subversive purposes.
Given that there’s no contemporary record (in the Gospels) of the disposition of The Cross, I’d guess that the Romans destroyed it quickly following the execution. Unless Pilate, or a Roman military unit decided to keep it as a momento/conversation piece.
* Thrown away
or
* Put back in the pile for the next load of crucifixions
or
* My preferred theory - it was proudly displayed by Pontius Pilate at the crossroads for a week to display his anti-Sanhedrin message about the King of the Jews.
The Romans could barely be persuaded to put a guard on Christs body - why would they guard the Cross? The Cross wasn't a battle-standard. It was many long years before it became a symbol of Christianity.
There's no evidence - as in none whatsoever - that the Cross was destroyed. It's certainly possible that pieces of it exist and have been preserved to this day