The earliest cited uses of this sense of gone in Green's Dictionary of Slang are actually from before the founding of the United States, even as early as the 16th century, so while it might not be called a "Britishism," it certainly originated in Britain and spread to use in the U.S. only later.
The first attested use is from Shakespeare.
c.1595 - [UK] - Shakespeare Love’s Labour’s Lost V ii: The party is gone; fellow Hector, she is gone; she is two months on her
way [...] the child brags in her belly already.
1672 - [UK] - Wycherley Love in a Wood Act V: Indeed, I found myself six months gone with Child, and saw no hopes of your getting me
1751 - [UK] - Smollett Peregrine Pickle (1964) 462: I was six months gone with child.
Later uses in American English appear, first attested by GDoS in 1933.
- 1933 [US] J.T. Farrell Gas-House McGinty 186: The wife’s two months gone.
The phrase also appears in the 1958 novella Breakfast at Tiffany's by notable American author Truman Capote.
But, after all, he knows I’m preggers. Well, I am, darling. Six weeks gone.
- Breakfast at Tiffany’s 75