As most researchers know history is not always correct without documentation. I have found this on Sarah Walker:
NameSara WalkerGenderFemaleBaptism Date01 Mar 1618Baptism PlaceRadcliffe,Lancashire,EnglandFatherJames WalkerFHL Film Number0547193, 0844812Household Members
Records also show that James and Elizabeth Walker were married in Radlliffe. Birth record also shows that James Jr. was born in Radcliffe as well. Nothing has been found on Phillip. Also, James Walker Sr. did not come to America and is not buried here. Most likely Phillip never knew his father as he was born the same year James died. Hense, "widow Walker" and following her two eldest children to America.
I have also found this:
11 Dec 1594 • Clitheroe, Lancashire, England
Baptism: 11 Dec 1594 St Mary Magdalene, Clitheroe, Lancashire, England Elizabeth Bayley - Supposed daughter of John Browne & Margaret Bayley Abode: Lynne Register: Baptisms 1569 - 1614, Page 1, Entry 326 Source: LDS Film 1278857
There has been difficulty in finding that James Walker married Elizabeth Browne other than the marriage record in Radcliffe. It may be possible that she was not given the last name of her supposed father "John Browne" but carried the name of her mother Margaret Bayley.