Step 1: Identify the key figures.
The question connects two important people from the history of Hyderabad: Salar Jung III and a Nizam (the ruler of the princely state of Hyderabad). The question also mentions the famous Salar Jung Museum.
Step 2: Understand who Salar Jung III was.
Nawab Mir Yousuf Ali Khan, known as Salar Jung III, was an aristocrat and a passionate collector of art and antiques from around the world. He served as the Prime Minister (or Diwan) of the princely state of Hyderabad from 1912 to 1914.
Step 3: Identify the ruler during Salar Jung III's tenure.
Salar Jung III served under Mir Osman Ali Khan, who was the seventh and the last Nizam of Hyderabad. He ruled from 1911 until 1948. Salar Jung III's tenure as Prime Minister was during the early years of Mir Osman Ali Khan's reign.
Step 4: Connect to the museum.
After his tenure, Salar Jung III dedicated his life to his passion for collecting. His vast private collection is what now constitutes the Salar Jung Museum in Hyderabad, which is one of India's three National Museums. The collection was gifted to the nation after his death. Therefore, Salar Jung III was the prime minister of Nizam Mir Osman Ali Khan.