There is a fragment of a concerto by WAM, from latter Salzburg years, I
seem to recall.  But it's really only fragmentary, not enough for someone
to complete.

Mozart also wrote a substantial portion of an Andantino, which HAS been
completed by several editors over the years.  I believe this is from
about 1780.  It's a lovely little piece (I've performed it), and the
cello writing is fairly idiomatic.

The fact that it wasn't completed speaks to your question: Mozart wrote
on demand in most cases (the "Haydn" string quartets are a famous example
of an exception), and he seems not to have had a particular soloist for
whom he would produce such a piece. The flute concerti, for example,
were written to curry favor with a member of the French aristocracy;
the oboe concerto alone, I think, was written out of admiration for the
soloist, Ramm, who was a member of the celebrated Mannheim orchestra--and
even in this case, the production of the concerto was a solid political
move. (I'll have to read up again on the horn concerti; these too may
have been primarily written out of friendship for Leutgeb.)

There were, sadly, no notable cellists in Mozart's circle.  Finally,
he did get commissions for the string quartets for the Prussian court.
Friedrich Wilhelm II was a fine cellist, and always seeking new pieces
featuring his instrument--hence the lovely writing for the cello in the
last 3 string quartets.

Cheers,

Roger Lebow
Sierra Madre, California