In 1938 Nicky answers his phone at home and straight away speaks to his friend calling from Czechoslovakia. At that time all trunk (long distance) and international calls to and from the UK were operator-connected and there would have been a delay before the two parties could speak to each other.
Nicholas Winton is seen sorting through coins. It is clearly visible that the British coins he is sorting are 2008 coins, featuring the Royal Coat of Arms design. These would not have existed in 1987, when this segment is set.
The Saab driven by Winton in the film's late-1980s sequences has registration plates made with an updated version of the "Charles Wright" typeface that only came into use in 2001.
On arrival of the first Kindertransport at Liverpool Street station (in 1938), the announcer refers to London Liverpool Street, as does a sign on the platform. However, London station names were not prefixed with 'London' until about 2007.
When they listen to the answer machine, for the library, the number given is 01636 but in 1987 there would be no 1. These were added in 1994/5.