Some more information here. I manually edited the Laravel code on the server, so that the command run looked like this;
sh -c ('/usr/bin/php' 'artisan' mycommand:command > '/dev/null' 2>&1 ; '/usr/bin/php' 'artisan' schedule:finish "framework/schedule-a839ecc42aed78e0646ec2cc1c7c1f0b6273d64b" "$?") > '/var/log/testlog.log' 2>&1 &
And the contents of testlog.log after a deploy are; "Could not open input file: artisan"
So it seems like whatever directory the command is initially run in, doesn't exist at the end. Possibly due to a sym link switch or something like this. I'll need to investigate further.