Are there really "pedestrians prohibited" signs (TSRGD diagram 625.1) at all entrances to this section of Titnore Lane, together with a traffic order implementing this?

Access tags like foot=no in OpenStreetMap are intended only to record the legal situation, not personal opinions on safety or convenience. If you wish to reduce the likelihood that pedestrian routers would use this route, adding the 60mph National Speed Limit might have been more useful.

https://wiki.openstreetmap.org/wiki/Key:access

The section of Titnore Lane between the access road to Swallows' Return and the pedestrian crossing island SW of St. Barnabas House has a pavement/sidewalk, so there is clearly no pedestrian prohibition here. I have updated this section in https://www.openstreetmap.org/changeset/157408024