Fascinating information!  Thanks to:
Nancy Ellen Chapman Crayton, grand daughter of Nancy Louise Wham b. August 6, 1892

I relied on two books – South Carolina Naturalizations 1783-1850 by Brent Holcomb which has the entry from Chester District (or County), SC  (Naturalized Citizens, Volume A 1802-1832)  According to Brent Holcomb, these papers are available through the Chester Co Clerk of Court’s Office.  Brent stated that he knew his book was probably not a complete record but these were all he found at that time.  His book only includes Joseph Wham who arrived in Chester County in 1807.

It is interesting that Benjamin and William filed their reports as required in 1812 but Joseph Wham apparently did not report but petitioned for citizenship in November 1813.  So we have Benjamin and William in Kenneth Scott’s book and Joseph Wham in Brent Holcomb’s book.  Curious.  Another interesting point about naturalization back then was that if the head of the family was granted citizenship, it extended to his wife and all minor children in the household.

Benjamin and William Wham’s petition for citizenship is recorded in Kenneth Scott’s book British Aliens in the United States During the War of 1812.  He transcribed from microfilm.  The records were gathered because British citizens were considered enemy aliens once the War of 1812 began and a notice was sent out that they were to report to the marshal of the state or territory “the persons composing their families, the places of their residence and their occupations or pursuits, and whether, and at what time, they have made the application to the courts required by law, as preparatory to their naturalization”.  So this report was more helpful in that it stated that Benjamin Wham’s family consisted of only himself at the time of the report Aug 31 – Sep 8, 1812.   But apparently his wife had recently died as he appears to be married on the 1810 Chester County Census.  The age range for him and his wife were between 45 years and older.  His petition in 1812 indicated that he was then 68 years old.  William Wham was 35 years old in 1812 and had eight people in his family.  The 1810 census shows himself and wife, plus two boys and four girls.

I checked and it appears that James Woodside filed a report between Sept 10, 1812 and Oct 17, 1812.  He is “an Irishman”, aged 47, living in the U.S. five years, married in the U.S. with 2 in the family.  Living in Greenville County, SC.  Curiously, on the 1810 Census he shows up with 5 children under 10 yrs, and 3 children between 10 and 16.  Maybe he married a widow with children who were born in and therefore wouldn't be included in the report?  He is a shoemaker so maybe he has apprentices?  Who knows.  But since he is older than both Joseph and William Wham it appears that Widow Woodside would have been the wife on the census with Benjamin in 1810.  But that’s just speculation, not proof.  I have found no documents so relating to Benjamin’s wife.