One of the children asked this question today. A little bit of research later... the answer is... yes, there was a Roman road that ran in the same direction as the Fairmile but it probably ran next to the course of the modern road rather than directly underneath it. It went through where Bix and Nettlebed are today and went all the way to Dorchester on Thames. Nobody knows the exact location of it though.
Maybe there's a Roman road under our school!