The first film shown on British television (the BBC) was 'Tell Me If it Hurts' (1934), featuring Peter Copley who can be seen in a BBC production this year (Riot at the Rites). This was a short film and was broadcast on 14th March 1938. It was screened on Channel 4 in the 80s.
The first feature film to be broadcast in full was shown by the BBC the same year, August 14th. This was The Student of Prague starring Anton Walbrook.