The Kings Cliffe line closed to passengers (Peterborough-Rugby service) in 1966, not sure exactly when it was dismantled but it was gone when I moved to the area in 1969, as I remember walking the trackbed from Fineshade to Kings Cliffe. You can't do that now as Fineshade woods have reclaimed it making it mostly impenetrable. The section from Nassington quarries to Peterborough remained in use into the 1970s, but I have no idea when it actually closed. It is now cut back to near Yarwell junction, and is now the Nene Valley heritage railway.
The Medbourne line was used by a LNWR pick-up goods as well as the GNR passenger service. It was originally double track, but was singled by 1905 when the Peterborough bound line was removed, the station building was on the Leicester bound line. It burned down shortly after closure in 1916. I have a photo taken from the B664 overbridge looking down at the station just after closure, there is a long line of wagons, all look like 5 plank opens, but due to the angle I can't make out any company markings.