Judging by the position of the shaft from Google Maps, that bend into a horizontal tunnel is not for running rails. The running rails would have been under the Strand roadway itself, some 40-50m from the vertical shaft..
This is east of Aldwych, where the diagram shows the line to be running parallel to, but about 100 yards to the south of, Fleet Street. (It passes just to the south of St Brides Church). The extract does not extend as far as Bouverie Street but it looks like the running tunnels are indeed in line with the north end of Temple Lane, where the shaft is.
The TfL Property Asset Register shows that the route of the running tunnels would have been just north of the northern end of Temple Lane, with some kind of cross-passage planned to connect to the shaft.