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)