Born in Hobart, Tasmania, and educated at Ogilvie High School, Neilson became a clerk, and married Jill Benjamin, daughter of Phyllis Benjamin, in Melbourne in 1948, they had one son Andrew and three daughters, Christine, Carol and Robin.
Neilson was elected to the Tasmanian House of Assembly on 23 November 1946 at the age of 21. For many years he was not only the youngest MHA to be elected in Tasmania, but also the youngest person to be elected to any Australian parliament.
He held various ministerial portfolios, from Tourism in 1956 to Attorney-General. He was Minister for Education for over ten years from 1958 to 1969, and again from 1972 to 1974. He was made Attorney-General again on 12 April 1974, and five days later was also made Deputy Premier, Police Minister and Environment Minister. When the Premier, Eric Reece was forced to retire due to his age, Neilson was elected Tasmanian leader of the ALP, and Premier of Tasmania on 31 March 1975. He resigned as premier, and from parliament, on 1 December 1977.