As far as I know, the Pegu Club Cocktail has no relation to Winston Churchill, but as it was popular during the 1920s and 1930s, it's not improbable that he sampled a few.
The Pegu Club Cocktail was first mentioned in the 1930 Savoy Cocktail Book. Harry Craddock credits the drink to the Pegu Club, a British officers establishment, in Burma. The drink traveled throughout the world and was a hit until World War II. As always, Esquire has the most amusing take on the drink's history.
There are variations on the recipe. I've tinkered with a few, and this is the one that I like best:
The Pegu Club Cocktail
Shake well with cracked ice, then strain into a chilled cocktail glass.
2 oz London dry gin (I personally like Plymouth or Hendrick's)
3/4 ounce orange curacao (or Cointreau, Gran Marnier or even Rhum Clement Creole Shrubb)
3/4 ounce lime juice
1 dash Angostura bitters
1 dash orange bitters (Optional. If you read any cocktail blogs, you'll see this ingredient's inclusion in the drink is fiercely debated. No, I'm not kidding. Cocktail aficionados are deadly serious about their tipple.)