Perhaps confusion begins with the name of the Ecclesiastical Parish: "Ditton Wood St. Mary" https://ukga.org/index.php?pageid=853
I think the town was never named Wood Ditton as evidenced below:
* Woodditton: https://www.british-history.ac.uk/vch/cambs/vol10/pp78-86
* Woodditton: https://www.google.com/maps/dir/Woodditton,+Newmarket,+UK//@52.1961...
* Woodditton: https://en.wikipedia.org/wiki/Woodditton