A search on both newspapers.com and World Cat of any of the various spellings of Billy or Plunkett (in quotes) that I could think of came up with nothing of use.

One thing I found which could be of importance is the British cartoonist Wally Fawkes. Though I haven't found a cartoon he made about jazz, and his name obviously differs greatly, he was active in the times you mentioned, British, and played in a jazz band himself. See some examples of his work here