Synonyms of speedily in English:
you should ensure that complaints are handled speedily
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;
informalp.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
North American informallickety-split
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