Mary's second marriage is on the LDS pilot site
http://search.labs.familysearch.orgThomas GOLDTHORPE, son of George GOLDTHORPE married Mary SCARGILL, daughter of Jonathan HALLAS.
Hopefully the father's name might be helpful in searching for a baptism, although I can't spot anything likely online at the moment.

The GAWTHORP family you found in 1861 looks a fair match to me - the children Joseph, Emma and Mary Ann match well with the SCARGILL children in 1851, and Charles (aged 10 in 1861) must have been born after the 1851 census and shortly before or after Joseph's death. (It isn't unusual to find children from a previous marriage recorded under their step-father's surname.)
I think this must be Mary with her son Charles in 1871, although strangely she is listed as SCARGILL:
RG10/4590/40/16 - Bottoms, HeckmondwikeMary SCARGILL - Head - Wid - Emley
Charles SCARGILL - Son - 20 - Coal Miner - Emley
Ellen SCARGILL - Wife - 20 - Dewsbury
John RANGLEY - Grandson - 7 - Scholar - Heckmondwike
There is a death for a Mary SCARGILL aged 66 in Wakefield registration district (which covers Heckmondwike) in 1874, but it seems strange that she would revert to her first married name.

Rambler