The Peter Atherton you refer to was undertood to be a clock maker by trade and it was he that built the 1st spinning frame as designed by Richard Arkwright.
Peter is further recorded as the manager of a cotton mill in Holywell until 1790.
Per History of greenfield valley by Davies & Williams
Returning to Burtonwood Chapelry , you should also be considering people living within the Parish of Winwick and the Chapelry of Farnworth (within the old Parish of Prescot) and all of these 3 Churches are on the I G I.
This area is full of Athertons and I think that you will find that "all roads lead you back to Ashton in Makerfield once in the Parish of Winwick".