Born in South Africa he moved to Britain in 1859 to continue his education and in 1864 began working with Peter W. Barlow and then spent time (c.1867) as assistant engineer on the Midland Railway between Bedford and London (working with Barlow's brother, William Henry Barlow).
Soon after, in 1869, he rejoined Barlow and they began work on designs for the Tower Subway, a tunnel under the river Thames in central London. This began a long fascination with the challenges of tunnelling through soft, waterlogged soil, and today Greathead is particularly remembered for his pioneering work in relation to tunnelling shield techniques.
He was Chief Engineer on the City and South London Railway (now part of the Northern Line, opened in 1890), and, shortly before his death in Streatham, London, began work on the Central Line (opened 1900) with Sir Benjamin Baker. He was also a consultant in relation to the construction of the Blackwall Tunnel and the Waterloo and City Line, both completed after his death.