According to The American Heritage Idioms Dictionary the idiomatic expression On the nose probably derives from boxing, suggesting that the opponent’s nose was the target:
Exactly, precisely; especially, at the appointed time or estimated amount.
For example, The busload of students arrived at the museum at ten o'clock right on the nose , or He guessed the final score on the nose.
This term, like on the button, may come from boxing, where the opponent's nose is a highly desired target. [c. 1930]
The Dictionary of Word and Phrase Origins by
Martin Manser, 2018, has an alternative suggestion to its origin:
The expression on the nose, meaning 'exactly; at target point' is said to have originated in the early day of radio broadcasting. Directors in soundprooof control rooms made certain signals to the assistants performing the actual programme. Putting the forefingers along the nose meant that the programme was running precisely on time. Other signs included the director “sawing” his throat forcefully to mean “cut”.
Early usage example from The Phrase Finder
The New Yorker: Volume 23, Part 6, 1948:
I really thought Knapsack could win the race, so I sent a boy to put a bet on him. I told the boy to put twenty-five thousand on the nose on Knapsack.