The phrase "upon the resignation" is correct and usable in written English.
It is usually used to indicate when something happened after another event. For example: The company filled the position shortly upon the resignation of the previous manager.
Brown was appointed to his position, not elected, upon the resignation of Tony Blair.
He succeeded to the prime ministry upon the resignation of B.J. Vorster in 1978.
Upon the resignation of Noah H. Swayne in 1881, President Rutherford B. Hayes nominated Matthews to the U.S. Supreme Court.
Suzuki became prime minister upon the resignation of Koiso Kuniaki on April 5, 1945, four days after U.S. forces had landed on Okinawa.
As lieutenant governor, Mr. Huckabee became governor upon the resignation, in 1996, of Jim Guy Tucker, who was convicted of fraud and conspiracy in a Whitewater-related trial.
After serving as federal treasurer from 1921 to 1923, he became prime minister upon the resignation of William Morris Hughes, forming a coalition of the National and Country parties.
He succeeded to the Irish presidency in 1976 upon the abrupt resignation of Cearbhall Ó Dálaigh.
Adversaries of the mayor seized upon the second resignation to criticize Mr. Bloomberg's handling of the education department, saying the system's integrity had been undermined.
He was appointed interim captain on 11 June upon the sudden resignation of Shaun Udal.
Upon the announced resignation, NBA Commissioner Walter Kennedy announced that the league would conduct a hearing on the matter.
Under Kentucky's first constitution (1792), the Speaker of the Kentucky Senate became acting governor upon the death, resignation, or removal of the sitting governor from office, until a new election could be held.
