My husband is a Livermore with paper trail through male line back to Robert Livermore b abt 1603. His autosomal DNA links him to lots of other Livermore descendants including a descendant of your Thomas's brother William (1791).
But as his ancestors at that time were from a small area of the country where Livermores were plentiful, this could be a co-incidence and his actual genetic link is to a different ancestor. That was why I had done some previous research and was looking again at records I had already viewed.
Hello LizzieL: I've been filling in my family tree on Ancestry, including Thomas's siblings (the children of James Livermore and Eleanor Buttell/Buttle), and remembered your husband's link to a descendant of William Livermore b.1791. Since my DNA is now online, I don't suppose you've had a recent flag of a distant relation? It may have come as my online name of doordie or family name Fuger.
I've looked at my husband's matches, but he doesn't seem to have a match to you.
His link to a descendant of William Livermore (your Thomas's brother) might not be through the Livermore's but to one of his other branches in that area. Or the tree that his match has on line may not be accurate.
My husband also has a match to a descendant of William Livermore son of Mary Lucas (Livermore) and Peter Rumbold. As William was not the son of Thomas Livermore (who had been transported), this link cannot be through Livermore DNA. It's too easy to jump to conclusions, when you see an uncommon surname in a match's tree and think the link is there.
Many trees have William and Thomas's father James descended from Peter Livermore (b abt 1594) of little Thurlow, Suffolk. One of their sons, John, sailed on one of the early immigrant ships to America and settled in Watertown, Massachusetts. Whether this John really was the ancestor of all the Livermores who claim him is unknown. Maybe they found them in the book by Thwing, see link below.
I have not yet found a link with my husband's Livermores to Peter of Little Thurlow. The earliest I have is a Robert who married in Ashdon, Essex in 1628, so I guess he was born about 1600ish. Ashdon is about 10 miles from Lt Thurlow and 7 from Little Abington, so they may be related many generations ago.
https://archive.org/details/livermorefamilyo00thwi/page/n5/mode/2up