Answer:
Unlikely. Roman tradition at the time usually had the vertical pole already at the site of execution, the convicted person only carried the crosspiece from the prison to the site of execution, where his arms were nailed to the crosspiece. He was then lifted by the crosspiece and the crosspiece suddenly dropped vertically into an iron receptacle on the pole (this shock alone killed some victims) after which the legs were nailed to the pole. Both pole and crosspiece were usually reused for many executions.
This is probably how Christ was crucified.