I don't know how recently it has happened, but there are now Caribbean archives to be found on the internet listing abstracts of Nevis wills. There is one relating to Thomas William dated 20 September 1766. The will indicates that Thomas Williams' niece, Martha Sanders was married to Thomas Wenham '"the elder". There is a record of the marriage between Martha and Thomas in Nevis on 1 Nov 1756.
I believe Thomas Wenham, the elder, and his wife Martha had numerous children; Martha Williams Wenham, Frances Wenham, Margaret Wenham, Mary Wenham and, presumably, Thomas Wenham.
In Thomas Williams' will it is mentioned that he "rented part of a plantation of [Thomas Wenham the elder] in island of Antiqua" called Budgeons Plantation.
Thomas Wenham is bequeathed "in fee piece of land, whereon a small square room now stands, in Charles Town, Nevis of about 63sq. ft, which was late the property of the sd. William Woodward."
A picture of the 2-storey Wenham House in Nevis is available online. Simply type in Thomas Wenham of Nevis. The house I understand no longer exists.
William Woolward was undoubtedly related to Fanny Woolward who I believe married Horatio Nelson in Nevis.
The Collins, mentioned in your blog, had thought there was a connection to Rev. George Wenham of Rotherfield (died 1736). There probably was a connection since it was George's branch of the family that was mentioned in the Visitations of Sussex and who were entitled to use the Wenham coat of arms.