I'll need to give this some thought, but as a train-timer my test would be where the station to station start to stop time is less than a minute. In essence, the shortest 'fastest time'.
Example on the Whitchurch branch already quoted and St James to Moorfields can be under a minute.
Ok some ideas based on Railway Performance Society station-to-station 'fastest times', many of which already mentioned
Start to stop one minute or less
James Street to Moorfields 23 chains. 0 minutes 45 seconds.
Canada Water to Rotherhithe. Nbd 0m 46s, sbd 0m 46s.
Farringdon to City Thameslink Sbd 0m 46s (nbd 0m 56s)
Blackfriars to City Thameslink. Nbd 0m 47s, (sbd 0m 50s)
Ty Glas - Birchgrove 19c. Nbd 0m 48s (sbd 0m 49s).
Stratford to Maryland ebd 0m 52s (wbd 1m 10s).
Alexandra Parade to Duke Street sbd 0m 49s. (nbd 0m 57s).
Rotherhithe to Wapping. Nbd 0m 52s (sbd 0m 55s).
Finchley Road & Frognal to West Hampstead. Ebd fastest 0m 55s (wbd fastest 0m 56s).
Clapham High St to Wandsworth Road 0m 57s.
Lelant Saltings to Lelant (37c). Fastest down 0m 56c, fastest up 1m 10s. (Lelant Saltings now closed).
Deansgate to Manchester Oxford Road to ebd 0m 58s (wbd 0m 59s).
Liverpool Lime Street to Liverpool Central (42c) 1m 00s
Close calls
Prestwick Town to Airport 1m 02s both directions
Exton to Lympstone Commando. Fastest down 1m 03c, fastest up 1m 03c.
Balloch Pier to Balloch Town nbd 1m 02s (sbd 1m 13s) Balloch Pier now closed.
Culrain to Invershin. Fastest down 1m 11s, fastest up 1m 04s.
Glasgow Queen St to High Street. Fastest down 1m 06s, fastest up 1m 04s.
Getting late, no doubt I have missed quite a few