Now, the good thing about that is that they are all available online from several archive sites. I will link to them from here, but you can find them on the Internet Archive as well. (My links are single MP3 files, though you can probably find other formats if you poke around.)
Around The World In 80 Days (23 October 1938)
A Christmas Carol (23 December 1938)
Dracula (11 July 1938)
The War Of The Worlds (30 October 1938)
Unfortunately no recording is known to survive of the fifth nominee, the BBC's 11 February 1938 live 38-minute TV adaptation of R.U.R. by Karel Čapek, thought to be the first ever BBC science fiction play.