Why wouldn't his followers and admirers cut a piece from it on Good Friday? Or the day after?
It was covered with Christ's sanctifying blood and every sliver would have been precious.
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.