Not a British example, but I think there was a line in Ireland that was built to the Irish 'standard' gauge of 5' 3" but wasn't connected to anything else.
The line you are thinking of is - I think - the Waterford and Tramore Railway, which ran from Waterford Manor Station (on the other side of the city to the main - now Plunkett - station) Tramore. It remained isolated until closure in the 1950s, even after being taken over by the Great Southern Rly, and so CIE. These did undertake exercises to bring in new rolling stock by road. Am I right in thinking that - as it had no other stations - Waterford Manor and Tramore were the only stations on a public passenger railway (not a museum or amusement operation), from which there was only rail connection to one other (the other of the pair)?
I sometimes think of the W&T as the simplest possible example of a railway in the British Isles, as a line linking two stations, and that being it...
There is no NR/LU connection at Upminster, the work was never completed.
There is a connection at Barking.
Until the LT and BR (as they then were) tracks were segregated from about 1960 (I'm not sure how long it took for complete speration to occur), the east end of District Line (beyond Campbell Road Junction (a bit east of Bow Road) was over the LTSR/Midland/LMSR, with District trains running by agreement to provide part of the service for that section (for many of the stations, meant all of the trains), and some through services from Southend onto the MDR and through to Ealing. Only after nationalisation under the BTC in 1948/49 was there any transfer from what had become the Railway Executive to the London Transport Executive.
One result of this separation was the reversal of the Upminster-Romford branch - it changed being one from the LT&S at Upminster (with through srvices from Southend, Tilbury and Grays) to Romford, to one from the GE at Romford to Upminster.