↑Source:Willard memoir : or, life and times of Major Simon Willard ; with notices of three generations of his descendants, and two collateral branches in the United States, has extensive lists of baptisms in Horsmonden of various Willards on pp. 39-40. The list includes both Simon and his first wife Mary Sharpe, but no children of Simon are included anywhere in this list. In the chapter Life of Simon Willard, p. 134, it says Simon embarked from England in April 1634, pointing out his wife was just 20 at the time, and p. 107, he was at Cambridge in the summer of 1634. In Sept. 1635, the right to settle at Concord was granted. His son Simon was the first of his own children that he recorded as clerk in Concord in 1639. Regarding Elizabeth's birth, this source says it is "uncertain whether born in Old or New England", but clearly if a birth date of mid-1634 or later is accepted, it would be New England.
According to new research published in 2016, Simon Willard's first wife was Mary Sharpee, baptized in 1609. See the mother's person page. Mary (Sharpee) Willard was 25, not 20, at the time of emigration to New England.