He was born in England. We know this from a letter which survives, dated 1221-1222, in which he directly states he was born in England - "nos omnes naturales homines Angliae" This is the only record or indication of his birth place.
The identity of his mother "Countess Ida" was one of the great genealogical puzzles which has only fairly recently been solved. Among the many theories of the past was various Ida's from continental families which is probably where the birth place of France came from.
The identification of his mother as Ida de Toeni, wife of Roger le Bigod, Earl of Norfolk has only recently been made. With his mother now known to be English, it is another strong indicator that he was born in England. It cannot be known that he was born at Woodstock manor, or any specific place in England.
The American Genealogist, volume 77 (April 2002): 137–149. Countess Ida, Mother of William Longespée, Illegitimate Son of Henry II, by Paul C. Reed.
The American Genealogist, volume 77 (October 2002): 279–281. William Longespée, Ralph Bigod, and Countess Ida, by Raymond W Phair.