Answer by terdon for Append current timestamp on files name from background running command

Whether the command is backgrounded or not is irrelevant, all you need to do is add the output of a date call to the file names you are creating. Using the exact same syntax as in your question, you could do:

sshpass -p prakash12 ssh -t -p $1 prakash@localhost \
    './test_new_update_script.sh > /home/log/unit_update_output.$(date '+%F-%T').log \
     2> /home/log/unit_update_error.$(date '+%F-%T').log < /dev/null | 
     echo $! > /home/log/unit_update_pids$(date '+%F-%T') &'

The trick is adding $(date '+%F-%T')., this will return (for example):

$ date '+%F-%T'
2014-04-08-18:54:52

So, the command above will create:

/home/log/unit_update_output.2014-04-08-18:57:02.log 
/home/log/unit_update_error.2014-04-08-18:57:02.log 
/home/log/unit_update_pids2014-04-08-18:57:02

See man date for the different formats you can have.