See Nine Hundred Years of Thweatt Bloodline family History by Silas Allen Thweatt (I have the book, but it’s online at http://web.comporium.net/~jaft/900.txt).
You may also get something out of "The Thweatt Family" by Claiborne Thweatt Smith, Jr. in The Southside Virginian (See http://archive.org/stream/southsidevirgini71989/southsidevirgini71989_djvu.txt).
James Thweatt, b 1643, was the immigrant to the United States.
There are many variations of the name Thweatt, including Threatt (mostly those who went to South Carolina from Virginia) and Threet (others who went to Southern States and Wyoming) and Threewits and Threat and Thret. It seems to have been tied to forms of Thwaite in England from 1066 and Wm the Conqueror, and, according to Silas, Denmark!