rapidly, swiftly, quickly, fast, post-haste, at (full) speed, at the speed of light, at full tilt, as fast as one's legs can carry one, at a gallop

promptly, immediately, briskly

hastily, hurriedly, precipitately

informal p.d.q. (pretty damn quick), double quick, at a lick, hell for leather, pronto, at the double, a mile a minute, like the wind, like a bomb, at warp speed, like a bat out of hell, like a scalded cat, like the deuce, like nobody's business, like (greased) lightning, like a madman/madwoman

British informal like the clappers, at a rate of knots, like billy-o

North American informal lickety-split

literary apace