Dugdale's Visitations of Lancashire (1664) cites only one child, a son Roger, married three times. This line is Bradshaw of Pennimgton, and there are *five* others. Bradshaw of Bradshaw, Bradshaw of Darcy-Lever, Bradshagh (Bradshaw) of the Hagh, Bradshaw of Pendleton, and Bradshaw of Prisall. That they were all connected at some point is likely, as all six families used very similar arms. (Bradshaw of Bradshaw, having the simplest form, may be taken to be the senior/original branch.)
Note: "John William" Bradshaw is particularly unlikely, as middle names were -as I have reminded all of you time and time again - Not A Thing Yet.