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.
AL: Squeaky, get back out there and... and... look for thugs.
JARVIS: Thugs, your--?
AL: Am-scray, Squeaky. Get lost. (JARVIS EXITS RIGHT)
It shows up in Billboard, 15 April 1944:
NEW YORK, April 8--A society gal, breaking in a new role as secretary in one of the local agencies, informed her boss that a certain performer wanted to see him.
"I'm too busy," said the percenter. "Tell that jerk to get lost."
The socialite - secretary went back and told the actor, "Mr. So-and-So is busy. He says for you to get lost."
Snippets of I Never Left Home (1944) by Bob Hope (snippets can have incorrect metadata, but this seems correct):
If I picked a nice parlay of three or four out-of-line cracks to hand to the whole British press, the American Embassy might suggest I get lost.
And from the same book, perhaps literal, but perhaps a Bob Hope half-joke:
It was like breaking sticks to get lost. The population doesn't help you much, either. They want you to get lost. And stay lost. To them everybody on the road they don't know is a probable Nazi spy.
Another possible 1944 is in Best stories of modern Bengal, Volume 1 by Dilip K. Gupta:
"Then that's all right. Tell her to get lost again." Brindaban said, "To tell the truth, that's what Haripada also would prefer. Oh, the scandal! It'll just be complete if she turns a prostitute! Then Haripada won't be able to show his face in Calcutta! Even some of his friends will visit her!"
"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:
Let's get lost, lost in each other's arms
Let's get lost, let them send out alarms
And though they'll think us rather rude
Let's tell the world we're in that crazy mood
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.
Listen, would you please get lost for a few minutes? Please, I'll be right up.
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:
If Kimberly were to walk in tomorrow ... I'd tell him to get lost.
They say that book was published in London, but apparently it was originally published by Rinehart & Company in 1946.