In the 1810 census, Nathaniel is listed as a head of household in North Kingstown, RI.
In the 1820 census, Nathaniel is listed again as a head of household in North Kingstown, RI. The household consists of three free white males -- one age 16-18, one age 16-25, and one age 45+ (who I assume is Nathaniel).
Also in the same census and same town is George Munday (age 25-44) with a wife and two children under age 10, one male and one female.
I found two references to a Nathaniel Munday of North Kingstown as follows. In an estate document for James Church, a letter of Administration was given to Nathaniel Munday, merchant, of NK, dated 7 March 1808; also Nathanial Munday of NK, yeoman, was mentioned in a deed of William Gardner of Exeter, dated 1 Aug 1812. These two documents may refer to the same individual.
He died in Wickford 2 January 1827, age 72 (Newport Mercury 6 January 1827 and lost gravestone). "Capt. Nathaniel MUNDAY died Jan 2, 1827 aged 72 years."