Ashby married Elizabeth Helen Margaret Farries, who he met while they were working together on incineration techniques for measuring carbon in tissue. They had two children, Michael and Peter.
In 1938, Ashby became professor of botany at the University of Sydney, a post he held until 1946. Between 1944 and 1945, he was Scientific Counsellor to Moscow. From 1947 to 1950, he was chair of botany at the University of Manchester and from 1950 to 1959 president and vice-chancellor of Queen's University, Belfast. For the University of Cambridge, he was Master of the Clare College, Cambridge from 1959 to 1967 and vice-chancellor from 1967 to 1969. Between 1970 and 1973, he was chair of the Royal Commission on Environmental Pollution. Ashby was knighted in 1956, and was created a life peer as Baron Ashby, of Brandon in the County of Suffolk.