I think the oldest signalling still in use on the Underground is at Edgware Road, which was last resignalled in 1926 (some reports say it was decommissioned recently, but I understand they have jumped the gun as the press release only reported the replacement as having been given the go-ahead)
The 1926 resignalling by the Metropolitan Railway was designed to accommodate a new relief line between Kilburn and Edgware road, and the platform destination displays were thus made capable of displaying exotic destinations like Watford and Quainton Road.
Once the Met and LER were both incorporated into London Transport, a less ambitious relief plan was formulated, using the Bakerloo Line and missing out Edgware Road altogether.