The 22-year-old Albanian national was arrested on suspicion of murder in the city of Durres last week following an international extradition request.
He remains in custody pending further work as part of the extradition process.
Site supervisor Woodward, 47, was killed while working on the Pressman Mastermelts building in Hatton Garden, EC1.
He was employed by refurbishment specialist G3 Construction UK Ltd.
Initial reports indicated that he may have fallen from height, but a post-mortem examination gave cause of death as blunt force trauma to the head.