Immediately. The OED cites a use as early as 1662; the subsequent examples given suggest that in roughly 1900, the predominant form changed from two words (straight away) to one (straightaway), though both versions are still found.
Since straightaway has a 100 percent precise and unobjectionable American equivalent, to wit, right away, its quite frequent use nowadays by U.S. writers is an excellent gauge of their unwavering fondness for NOOBs.
“I knew straightaway what had gone wrong—caps lock was depressed by accident—but instead of simply taking my lumps and re-entering my password, I vented: ‘Is there anything on the computer keyboard more annoying than the caps lock key?'” (Mathew X.J. Malady [“a writer and editor living in Manhattan”], Slate.com. February 1, 2012)
“A comedy about business consultants? Get them to a strip club straight away (‘House of Lies’ pilot, last month).(Neil Genzlinger, New York Times, February 4, 2012)
notoneoffbritishisms.com 
Is it not more common for words frequently used together: straight away
to move toward hyphenation: straight-away
and then to a compound: straightaway?
“the subsequent examples given suggest that in roughly 1900, the predominant form changed from two words (straight away) to one (straightaway)”
A handful of OED citations can hardly be taken as statistically significant! Native-speaker intuition and corpus evidence suggest otherwise.
Never heard or read this as one word in the UK. On the other hand, I have heard it as a motor racing term from the US.
It’s like the Yankee habit of running together “under” and “way”. 😉
“Since straightaway has a 100 percent precise and unobjectionable American equivalent, to wit, right away, its quite frequent use nowadays by U.S. writers is an excellent gauge of their unwavering fondness for NOOBs.”
Well, not really. It’s more likely to be part of the inexhaustible appetite of writers the world over for synonyms and novelty, in an attempt to keep their composition fresh. The implication that Americans would resist a new word simply because they already have one with the same meaning is to reinforce prejudices about their insularity and conservatism. English is a world language, so why shouldn’t speakers of it freely interchange vocabulary and contribute to its already impressive wealth of expressions?
Over here “right away” is used as a synonym for “straightaway” so we don’t have to keep repeating the same word – just as we revived “maybe” from the U.S. to avoid having to keep saying “perhaps”.
Well, Harry, I am sometimes known as Ben “Can I qualify that for you” Yagoda, so I certainly did not and would not claim the OED’s examples were statistically significant. I said the examples “suggest” a change from two-word to one-word predominance; and I neglected to say that the OED itself lists the one-word version, noting “Also (esp. in earlier use) written as two words.” But from yours and others’ comments, I accept that OED is wrong and one word is, or appears to be, still more common.
And David, I completely agree with you about writers’ appetite for synonyms and novelty, but the point is that some synonyms and novelty are more equal than others. At this particular point, for some reason, American scribes (novelty!) are sweet on Britishisms.
.
“De gustibus non est disputandum.”
There’s no accounting for taste?
One more thought on the two words/one word issue. One-word “straightaway” is commonly found in coverage of various racing sports (“the straightaway”) and of American baseball (“hit the ball to straightaway center field”). As synonym for “immediately,” my hunch is that it’s usually one word before the verb (“I’ll straightaway get this sorted”) and could go either way after the verb (I’ll get it sorted straight away”). That is all.
Dave –
“There’s no use disputing matters of taste.” – like choice of words.
As if a word had a meaning out of context.
“Straightway” (two syllables) is found quite frequently in the King James Bible.
Is it me or do Americans tend to use “instantaneously” to mean “straight away”, as in “I’ll do that instantaneously”?
Don’t they say “momentarily” – which means something rather different to us?
Straight away would have the emphasis on the first and third syllables, the same as it is spoken, whereas straightaway would have the emphasis on the first syllable only, not as it is spoken,