Academic career
Robertson unsuccessfully applied to be made a fellow of Trinity in 1936: he was required to submit written work in support of his application, and did so on the material from Ithaca, but his examiner mistakenly sent Robertson's file to the Oxford art historian John Beazley, who was supporting Dale Trendall's competing application. Robertson instead took a post in September 1936 as Assistant Keeper in the Greek and Roman department of the British Museum, cataloguing the pottery from the excavations at Al-Mina in Syria led by Leonard Woolley in 1936–1937. In 1937–1938, staff in the department carried out an aggressive cleaning of the Elgin Marbles, using copper tools and abrasive silicon carbide, at the request of the entrepreneur Joseph Duveen; Duveen wanted the sculptures, originally painted, to look whiter for their display in a new gallery which he was funding. Three members of staff left the museum's employment as a result; Robertson was not involved, and so kept his job (leaving him as the only member of the department). He was, however, demoted in seniority. As a consequence of the departures, Denys Haynes was recruited as an Assistant Keeper, and the art historian Bernard Ashmole, the Yates Professor of Classical Art and Archaeology at University College London, was brought in on a part-time basis to run the Department of Greek and Roman Antiquities. Haynes and Robertson became lifelong friends, and Robertson later wrote of the "precious apprenticeship" he gained from working with Ashmole.
Robertson attended a classical conference in Berlin in August 1939, on behalf of the British Museum; he was recalled shortly before the outbreak of the Second World War. From 23 August, he took part in the removal of material from the museum to London Underground stations and country houses, so as to protect the artefacts from bombing. In 1940, he enlisted in the British Army as a member of the Royal Signals, but was soon transferred to the Intelligence Corps and trained to work in cryptanalysis at Bletchley Park. Sparkes later wrote that his military service was largely unsuccessful and characterised by "mind-numbing boredom". In late 1942, shortly after the end of the Second Battle of El Alamein in November, he was moved to the Secret Intelligence Service and sent to Cairo to work with Alan Wace, a fellow archaeologist and former director of the BSA, who had requested Robertson's assistance in carrying out intelligence work in the British embassy there. Due to an administrative mistake, Robertson was transferred to Naples rather than to Athens in 1943–1944, and later served in Salonica in Greece alongside Philby.
Robertson left military service in 1946, and returned to the British Museum, where he assisted in returning the evacuated collections to the galleries. He resigned in 1948 to succeed Ashmole as Yates Professor at UCL. During his tenure in London, he began to grow a beard, but was ordered to stop by his superiors: his obituarist Brian Sparkes wrote that they were concerned that he was displaying "arty" inclinations considered unbecoming of a professor. He was a visiting fellow of the BSA for the 1957–1958 academic year. He published his first book, Greek Painting, in 1959. It used the surviving Greek paintings on vases and other artistic works to reconstruct now-lost frescoes described by ancient authors. Between 1959 and 1968, he was chair of the governing council of the BSA.
In 1961, Robertson again succeeded Ashmole, this time as Lincoln Professor of Classical Archaeology and Art at the University of Oxford, in which role he served until his retirement in 1978. He edited the second volume of the BSA's excavations at Perachora in 1962, following the deaths of Payne (who had conducted the excavations) and of the editors initially appointed to publish the work. In 1968–1969, he was a visiting scholar at the Institute for Advanced Study in Princeton, New Jersey; he also held a visiting appointment at the J. Paul Getty Museum in Los Angeles in 1980. In 1992, he published The Art of Vase-Painting in Classical Athens, continuing and concluding his scholarly interest in Athenian painting.
From 1994, Robertson suffered from Guillain–Barré syndrome and myasthenia gravis. He died of cardiovascular disease and bronchopneumonia at home in Cambridge on 26 December 2004, and was buried in Cambridge City Cemetery.