I am not sure where that information came from, but in the book, Visitations to Glocestershire 1635, it shows the whole lineage back to Henry Willes, there may be some problems with spelling. Some decendants on the list spelled Guilliam or Guillim or Guillum or Gillim or even Gwilliam. There is also a problem with Isobell Dean being spelled Ysabel, Iylbell and otheres along with the last name, is is Dean or Deane. The book shows Deane. I found Several John Deanes about the same period, one of whom seem to get into a lot of trouble with Elizabeth the 1st. But one thing I have noted that according to the Visitations Book, Henry Willes wife, Jane or Janne, daughter of Lawrance, is listed in the geneology as surname Wins. but the visitation was misread and does not say Wins but says W "superscript"MS. I am not sure what this means. They used Sr to denote Sir and also St to denote Saint. so what is Wms., There are some Henry Wiles in Wales, so I am tracing that down and will post here.

I went back to the original Visitations of Gloucestershire in 1623. It seems that the LDS record was wrong, and so a lot of information that is on ancestry.com is wrong. It shows the parents of Thomas Guilliam as Henry Willes and Janne d. Lawrance Wins. But on the original document it cleary shows an abbreviation WMS. (not Wins). Looking in the index a the back of the book, it Lists, Jane (not Janne) and Lawrence WILLIAMS to be listed on page 251. 251 is the page that the Guillims are listed.

WMS was simply a abbreviation for Williams. The time of Thomas Guillims birth was near the time of Edward the 1st, (Longshanks), IT WAS REquired that all children born of royalty took the name of the royal line, that explains why Henry Wills son was name Thomas Guillims, the Welsh name Williams.

Good reading is the Diary of Mrs. John Graves Simcoe, it details this name change from i.e. William son of Thomas to Thomas Williams.