Son of Richard Neville and Barbara Arden, Edmund 'Edward" Neville. Thomas Edward Neville Westmorland was born in 1565, to Edmund 'Edward' Westmorland (7th Earl of Wetmoreland) and Jane Smyth..
Edmund was born in 1555, in Upper Wick, Worcestershire, England.
Jane was born in 1574, in Comebridge, Staffordshire, England.
https://www.myheritage.com/names/thomas_westmorland"Thomas Neville Westmoreland, known as Lord Westmorland, followed in his father's footsteps, never used the name Neville, being known as Thomas Westmorland . This Thomas Westmoreland named his first son, James Westmorland and his second son, Thomas Westmorland. His son, James, departed England in 1652 at the age of 17 and sailed to America. He landed in Boston, Massachusetts, changed the spelling of his name to Westmoreland and continued south to the State of Virginia where he married and raised a family."
I don't doubt that your DNA matches the Neville name/Dna but I'm not sure about the Charles Neville/unknown woman parentage also Thomas Neville Westmoreland appears to have an older brother James Westmoreland who died after Thomas, so James children would have inherited the Lordship if he had any.
Charles Neville, 6th Earl of Westmorland was born between 18 August 1542 and 28 August 1543.1 He was the son of Henry Neville, 5th Earl of Westmorland and Lady Anne Manners. He married Lady Jane Howard, daughter of Henry Howard, Earl of Surrey and Lady Frances de Vere, between 1563 and 1564.2 He died on 16 November 1601 at Nieuport, Belgium, WITHOUT ANY SURVIVING MALE ISSUE.
http://www.thepeerage.com/p1417.htm#i14163If Charles Neville had a first wife, who was legal then you would think if would be a Catholic marriage and be well recorded.