In 1860, when the Burke and Wills Expedition departed Melbourne, the colony of Victoria had responsible government but still reported directly to the Imperial Governement in Westminster.
See also http://en.wikipedia.org/wiki/Politics_of_the_United_Kingdom.

Monarchy

Alexandrina Victoria, born 24 May 1819,
was Queen of the United Kingdom of Great Britain and Ireland from 20 June 1837,
and Empress of India from 1 January 1877,
until her death on 22 January 1901.

Prime Ministers
http://en.wikipedia.org/wiki/List_of_Prime_Ministers_of_the_United_Kingdom

Colonial Secretary / Secretary of State for the Colonies
http://www.nationmaster.com/encyclopedia/Colonial-secretary
http://en.wikipedia.org/wiki/Secretary_of_State_for_War_and_the_Colonies

The Colonial Secretary was the British Cabinet official in charge of managing the various British colonies. The position was first created in 1768 to deal with the increasingly troublesome North American colonies. Previously those responsibilities had fallen to the Secretary of State for the Southern Department, who was responsible for Southern England, Wales, Ireland, the American colonies, and relations with the Catholic and Muslim states of Europe.

In 1782, following the loss of the American colonies, the office was abolished, and its duties given to the Home Secretary. In 1794 a new office was created for Henry Dundas, the Secretary of State for War, who now took responsibility for the Colonies, and in 1801 the position was renamed " Secretary of State for War and the Colonies". The Secretary of State for War and the Colonies was a British cabinet level position responsible for the army and the British colonies (other than India). The Department was created in 1801. In 1854 military reforms led to the split into the separate offices of Secretary of State for War and Secretary of State for the Colonies.

First Lord of the Treasury
http://en.wikipedia.org/wiki/First_Lord_of_the_Treasury

The First Lord of the Treasury is the head of the commission exercising the ancient office of Lord High Treasurer in the United Kingdom, usually but not always the Prime Minister. Beginning in the 17th century, the Treasury was frequently entrusted to a commission, rather than to a single individual, and after 1714, it was always in commission. The commissioners were referred to as Lords Commissioners of the Treasury, and given a number based on seniority. Eventually, the First Lord of the Treasury came to be seen as the natural head of any ministry, and, from Robert Walpole on, began to be known, unofficially, as the prime minister.

Lord Priviy Seal
http://en.wikipedia.org/wiki/Lord_Privy_Seal

The Lord Privy Seal or Lord Keeper of the Privy Seal is one of the traditional sinecure offices in the British Cabinet. Originally, its holder was responsible for the monarch's personal ("privy") seal (as opposed to the Great Seal of State, which is in the care of the Lord Chancellor).