Astute reader John Wall points to a sentence in today’s New York Times: “The Guardian said on Monday that it intended to cut its costs sharply in an effort to reduce its losses and break even at an operating level in three years’ time.”
He notes, “I think of the phrase with ‘time’ preceded by the possessive form of a measure of time as a very British expression, as in ‘two hours’ time’ or, in this case ‘three years’ time.’ … A native USA speaker/writer of English would, I think, write ‘in three years,’ without the possessive apostrophe or the word ‘time.'”
I was not aware of the Britishness of the formulation, but John is correct, according to this Google Ngrams Viewer graph showing the relative frequency of the phrase “years time” in British and U.S. sources. (Ngram viewer doesn’t recognize apostrophes, so I left it out; and I didn’t include the number of years, so as to include a wider range of citations.)
Like a number of other NOOBs, “___ years’ time” wasn’t especially British until fairly recently–in this case, the 1920s. It steadily increased in popularity till the late 1980s–the moment when the modest American revival commenced.
notoneoffbritishisms.com 
It’s not only ‘x years’ time’ which features this construction. Another, similar, example is the phrase, ‘x weeks’ notice’.
A film with such a title was criticized for leaving out the apostrophe. I must agree that ‘Two Weeks Notice’ does, to me, look distinctly odd.
Surely the movie title should have been, ‘Two Weeks’ Notice’?
I would interpret the two expressions slightly differently. Taken literally, the expression with the apostrophe must be three years of time but I’d always read it that if the Guardian says it aims to to break even in three years’ time, that defines a point three years from now. If it says it aims to break even in three years, I’d take it that it may be leaving it open as to when it will do that: it may intend to specify the amount of time involved but not the start or end of the three-year period to achieve that particular goal.
For some reason I have noticing this a lot on British TV lately, and I think the direct U.S. equivalent would be “three years from now.”
The Guardian will not break even in 3 years’ time or even 10 years’ time