The initial impetus is generally thought to have been Hadrian’s, during his visit to Britain in AD 122, and that he put his governor, A. Platorius Nepos, in charge of the project (it is his name that appears on the building inscriptions from Milecastle 38). This would accord with the duties of a governor as outlined by the jurist Ulpian.
Since the construction work was divided between the three British legions, II Augusta, VI Victrix, and XX Valeria Victrix, it might be supposed that the legionary legates (or whoever was in charge of the respective detachments) had some input into the organisation of labour, but we are on firmer ground with how individual stints of work were supervised. Building stones are known from a number of places along the Wall recording work by gangs under a particular centurion, sometimes even citing a particular cohort of an unnamed legion, but not noting the distance constructed (unlike the Antonine Wall, where stints are measured). Literary sources, such as the writer Vegetius, record how fortification work undertaken by the Roman army was generally checked by a centurion using a ten-foot measuring rod.