Right away; immediately. (Thanks to Laura Zuccardy.) “And as if for extra emphasis, [Monica Seles] broke Huber straight away in the opening game of the second.” (New York Times, August 19, 1995)/”Oh, he apologized/then and brought me straight away/a new machete,/with which I sliced/the brisket….” (Bob Hicok, “A Night Out,” published in The New Yorker, May 31, 2010). Google Ngram.
notoneoffbritishisms.com 
Very common, in the form “straightway”, in the Authorized Version (“King James Bible”).