This is my working hypothesis - the way I see it as of this moment!!

William Bateman had immigrated to the Massachusetts Bay Colony by 18 Sept 1631 when he attended the inquest of another William Bateman's death in Charlestown. [Were they father and son?]  He had arrived with his wife and two or three children. 
1632 William was the defendant Sgt. William Bateman in a case 5 June
1634 The general court revoked William Bateman's freeman oath 14 May
1638 William was granted planting ground in Charlestown, upon settling title with the local Indian leaders
1640 William had moved to Concord where he took the freeman's oath 2 June
1641 William and son Thomas estate valued at £348 in Concord
1644 William moved to Fairfield, CT with Concord minister John Jones whose congregation went with him to Fairfield
1650 Bought land from Thomas Sherwood
1654 William sold his home lot to son-in-law Henry Lyon
1656/57  William wrote his will 24 March
1657  William Bateman died 27 April 1657 in Fairfield, CT     

Clues of William Bateman's origin: Concord was settled in 1635 by Watertown residents; he may have come from there. Or from Cambridge where other early Concord settlers originated.