Well, there's more than one way to do it (including using Perl, where that's the motto), but probably I'd write it like this:
find . -name '[A-Z]_*.txt' -type f -print0 |
xargs -0 modify_rename.sh
And then I'd write the script modify_rename.sh like this:
#!/bin/sh
for file in "$@"
do
dirname=$(dirname "$file")
basename=$(basename "$file" .txt)
leadname=${file%_*}
outname="$dirname/${leadname}_result.txt"
# Optionally check for pre-existence of $outname
{
# Optionally echo "$basename.txt" instead of "$file"
echo "$file"
# Does this invocation of CMD write to standard output?
# If not, adjust invocation appropriately.
CMD "$file"
} > "$outname"
done
The advantage of this separation into separate scripting operations is that the rename/modify operation can be checked out separately from the search process - which runs less risk of zapping your entire directory structure with bad commands.
Bash has the tools to avoid invoking basename and dirname but the notation is moderatly excruciating; I find the clarity of the command names worth having. I'd be happy if bash implemented them as built-ins. There are plenty of other ways to get the prefix of the file; this should be safe, though, even in the presence of spaces (tabs, newlines) in file or directory names because of the careful use of double quotes.