Hi Ruth: Since I am researching Kentucky Wilsons who moved to Rush Co., Indiana, your posting got my attention. I looked Thomas Oldham up on the 1820 Kentucky census and found him in Campbell Co., CovingtonTownship. The interesting thing is he is right next door to a Thomas Wilson. Both men are age 16-26, both of their wives are also 16-26, and they each have a son under age 10.
I decided to look at the 1810 Campbell, KY census to follow up on the Wilson family. There were three Wilson families: James Willson, John Willson, and William Willson. The James Willson did not have any daughters, but John and William each had daughters that would fit your Elizabeth's age -- each had a daughter under 10 and a daughter 10-16. Also, John had two sons age 16-18, and William had one son 10-16 and one son 16-18. So Thomas Wilson on the 1820 census would have to be one of those boys, with John or William as his father.
There was a Thomas Oldham in Guernsey Co., Ohio on the 1820 census, also. But he is 26-45 years of age and has at least 9 children. Doesn't seem likely.
My Wilsons were from Fleming Co. and possibly Bourbon and Mason Co., KY. It doesn't sound like we are from the same batch -- shucks. Hope some of this helps you in your search.