I have established that this term is an American idiom. Does anyone know when it came to be popular use or was first used there?
Get lost! dates from at least 1944 in popular media, and in speech is likely to pre-date this somewhat.
It shows up in Billboard, 15 April 1944:
Snippets of I Never Left Home (1944) by Bob Hope (snippets can have incorrect metadata, but this seems correct):
And from the same book, perhaps literal, but perhaps a Bob Hope half-joke:
Another possible 1944 is in Best stories of modern Bengal, Volume 1 by Dilip K. Gupta:
"Let's Get Lost", a torch ballad by Frank Loesser and Jimmy McHugh (sung here by Lina Romay, but also by Jimmy Dorsey, and by Mary Martin in the 1942 film Happy Go Lucky, and later by Frank Sinatra), was popular in 1943 and often in Billboard's top ten. It goes:
Although not directly using the imperative idiom, I think it's suggestive of it and likewise helped popularise it.
I take it that you mean the imperative "Get lost!" rather than the indicative "Americans get lost".
The earliest I found using Google was in Life Magazine on 13 October 1947.
There's not enough to be sure, but that may actually be a quote from Frederic Wakeman's The Hucksters, chapter 5, which OED also gives as a 1947 citation:
They say that book was published in London, but apparently it was originally published by Rinehart & Company in 1946.