During the Second World War, Robertson served briefly in the Royal Signals before being transferred to intelligence work, in which capacity he was a subordinate of the archaeologist Alan Wace and a colleague of the Soviet double agent Kim Philby.

His work on Greek art developed that of John Beazley, who had pioneered the study of Attic vase-painting in the first half of the twentieth century.

[3]: 557 [2]: 321  His childhood friends included Kim Philby, later a double agent for the Soviet union within the British Secret Intelligence Service.

[2]: 322 Robertson returned to England in 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.

[2]: 323 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.

[2]: 323  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.

In an obituary of Robertson, the archaeologist Brian Sparkes wrote that his military service was largely unsuccessful and characterised by "mind-numbing boredom".

[2]: 322  In 1959, he published his first book, Greek Painting, in which he used vase-paintings and work in other media to try to recreate the lost wall-paintings known only through textual references.

[2]: 322  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.

[6]: 461 Robertson's History of Greek Art, which first appeared in 1975, was still considered an authoritative text and used for its breadth of learning and deep understanding of the topic in the twenty-first century.

As a poet Robertson published various collections, including Crooked Connections (1970), For Rachel (1972), A Hot Bath at Bedtime (1975), and The Sleeping Beauty's Prince (1977).

[2]: 324  His paternal aunt, Agnes Arber, was a botanist: she was the third woman and the first female life scientist to be granted membership of the Royal Society.