Walter Monckton helped to write the speech but Churchill was with the King shortly before he broadcasted and added some little touches, notably the reference to the Duke of York (the new King) having 'the matchless blessings' of a wife and children, which he did not possess.*
(Monckton, btw, had been chief legal adviser to the Duchy of Cornwall since 1932 and was well known to the King. He was adviser to both Edward and Wallis throughout the Abdication crisis.)
*Not that there had been anything stopping Edward from marrying earlier, just a devotion to women who were already married, from 1917 onwards.