It sounds like "On The Beach" by Nevil Shute, which was originally published as a story called "The Last Days on Earth" in 1957.
From the Wikipedia article:
Global air currents are slowly carrying the lethal nuclear fallout across the Intertropical Convergence Zone to the Southern Hemisphere. The only parts of the planet still habitable are Australia, New Zealand, South Africa, and the southern parts of South America, although they are slowly succumbing to radiation poisoning as well. Life in Melbourne continues reasonably normally, although the near-complete lack of motor fuels makes traveling difficult.
The characters make their best efforts to enjoy what time remains to them, speaking of small pleasures and continuing their customary activities. The Holmeses plant a garden that they will never see; Moira initially acts as a socialite – drinking and partying excessively – but upon meeting Towers takes classes in typing and shorthand; Osborne and others organize a dangerous motor race that results in the violent deaths of several participants; elderly members of a "gentlemen's club" drink up the wine in the club's cellar, debate over whether to move the fishing season up, and fret about whether agriculturally destructive rabbits will survive human beings. Towers goes on a fishing trip with Davidson, but they do not become sexually involved, as he wants to remain loyal to his dead wife, a decision Moira accepts.
The novel is available at Faded Page, although I was unable to find a copy of the original short story to verify if it is the one you remember.
