Cockburn remained in the UK and was educated at Highgate School, and King's College London, he obtained the degree of M.D.

Between 1884 and 1888, during Cockburn's parliamentary career, the South Australian Railways line through Jamestown to Petersburg was extended to the border to meet the tramway built by the Silverton Tramway Company, which linked the growing mines of Broken Hill to the South Australian coast at Port Pirie, where a smelter was built in 1889, effectively capturing the economic benefits of the Broken Hill mining field for South Australia.

In 1884 Cockburn was able to pass progressive legislation including succession duties and land tax, and in 1886 was involved in introducing payment for members of the South Australian parliament.

On 27 June 1889, Cockburn became the first doctor to become premier, a role he held for fourteen months before he lost a no-confidence motion and handing back to Thomas Playford.

At the Convention he had the equal best attendance record of the fifty or so delegates, but almost as often as not was on the losing side in its votes, reflecting in part his independence of faction.

[citation needed] He continued to play a part in women's suffrage upon his return to London and, along with his wife, was active in the suffragette movement in England.