A quick search of Yahoo yielded 23 accessible sites that mention the song; only 9 name a composer: plain "Arne" once, Thomas gets two votes, Michael six; so he seems to be winning so far!  The Oxford Companion also says of Michael:

Born in London in 1740, and died there in 1786, aged forty-six.  He was the (natural) son of [Thomas Arne].  He was unsuccessful as an opera singer, ruined himself in attempting to find the Philosopher's Stone, conducted the first performance of Handel's Messiah ever heard in Germany (Hamburg, 1772 -thirty years after its composition!) and composed music for the stage -some of it whilst in prison for debt.

Malcolm