I actually remember Fats Waller playing live on the radio.  Fats was
as near as you could get to genuine jazz in wartime London.  Whenever
he visited the BBC used to organise some recordings, but rather than
jazz piano, he usually elected to play church music on organs.

As a schoolboy just beginning to take an interest in such things,
I thought this the most appalling and flagrant waste of resources (to
put the situation politely), as we had quite enough church music already
and it was Fats's secular music that sent me into transports of joy.

Happily, his reputation was such that as soon as materials were
available after the war HMV began to reissue his 1938 piano jazz.
I still have somewhere the 78 of "Alligator Crawl" that set me on the
jazz trail.

Dan Wilson, London