I think most of the “underground”, at least the older lines are operated by TFL. The TFL also operates some of the “overground” lines as a franchise owner. This line runs on national rail lines out to Reading so maybe it if more like “overground” than “underground”. I thought this project would fall under TFL regardless so it is just a joke really.

By the way, for national rail trains, private companies also own most of the rolling stock which the franchise owners lease. The whole British railway system is an absolute mess…