The story of the first printed Greek New Testament is really the story of two editions racing each other, only one of which is usually remembered as first. Cardinal Francisco Jimenez de Cisneros's team at Alcala de Henares in Spain had finished printing their Greek and Latin New Testament, part of the larger Complutensian Polyglot Bible project, by 1514, two years before Erasmus's edition appeared. But the Polyglot could not be published, in the sense of being distributed and sold, until it received papal approval, and that approval did not arrive until 1520, with the completed multi-volume Bible not actually reaching readers until 1522.
Erasmus and his publisher had no such wait to endure, and his edition reached the market in 1516, four years ahead of the technically earlier Spanish work. Because printed availability rather than printing date is what determined which text scholars, reformers and translators actually used, Erasmus's hastily assembled edition, not Cisneros's more careful one, became the Greek New Testament of the Reformation, the direct ancestor of the Textus Receptus that underlies Tyndale's English New Testament, Luther's German Bible, and eventually the King James Version. Being technically second did nothing to prevent Erasmus's edition from effectively winning.