This information provided in the link above was advanced by our archivist, Steve Boerner, and the following response is from one of our genealogists, Susan McMahon.
Thanks for sharing this interesting article! It seems that the carriage at Belle Meade Plantation and Museum probably was the one made for Francis Townsend Underhill by William T. Downs! I found information on the book you mentioned and there is a photo of a carriage that is attributed to the book and it looks just like the Tally Ho. The article mentions "Frank A. Underhill of Oyster Bay" but there is no Frank A. Underhill in the Oyster Bay area at the time. At least from what I could find.
Also, what is interesting is that William Thomas Downs was married to Helen R. Underhill a descendant of Capt. John's son David. They resided in Huntington. The 1870 and 1880 Census records show he was a "Carriage Maker". William (b 1848-d Nov 1884) was 36 yrs old when he died so, as the article mentions, he probably built only one Tally Ho carriage in his lifetime.