Your requirement if taken literally is an impractical one, because it is based in a slight misunderstanding. Fundamentally, to get the output to go in a file, you will have to change something to direct it there - which would violate your literal constraint.
But the practical problem is solvable, because unless explicitly counteracted in the child, the output directions configured in a parent process will be inherited. So you only have to setup the redirection once, using either a shell, or a custom launcher program or intermediary. After that it will be inherited.
So, for example:
cat > test.sh
#/bin/sh
echo "hello on stdout"
rm nosuchfile
./test2.sh
And a child script for it to call
cat > test2.sh
#/bin/sh
echo "hello on stdout from script 2"
rm thisfileisnteither
./nonexistantscript.sh
Run the first script redirecting both stdout and stderr (bash version - and you can do this in many ways such as by writing a C program that redirects its outputs then exec()'s your real program)
./test.sh &> logfile
Now examine the file and see results from stdout and stderr of both parent and child.
cat logfile
hello on stdout
rm: nosuchfile: No such file or directory
hello on stdout from script 2
rm: thisfileisnteither: No such file or directory
./test2.sh: line 4: ./nonexistantscript.sh: No such file or directory
Of course if you really dislike this, you can always always modify the kernel - but again, that is changing something (and a very ungainly solution too).