chdir($workdir) or die "Unable to enter dir $workdir:$!\n";
    ...
    foreach my $name (@names){
        ...
        if (-d $name){
            &ScanDirectory($name);     <-- This changes chdir.
            next;                      <-- Never changed back.
        }
        ...
    }
    ...
[download]

Fix:

use File::Spec::Functions qw( catfile );

# This subroutine takes the name of a directory and recursively scans
# down the filesystem from that point looking for files named "core"
sub ScanDirectory{
    my $workdir = shift;
    if ($workdir =~ /^[a-zA-Z]:\z/) {
       $workdir .= '.';   # Fix bug in Windows.
    }

    print "Work dir = $workdir\n";

    opendir(local *DIR, $workdir)
        or die "Unable to open $workdir: $!\n";
    my @names = readdir(DIR)
        or die "Unable to read $workdir: $!\n";
    closedir(DIR);

    foreach my $name (@names){
        next if ($name eq ".");
        next if ($name eq "..");
        my $fqname = catfile($workdir, $name);

        if (-d $fqname){                  # is this a directory?
            ScanDirectory($fqname);
            next;
        }

        print "Name = $name\n";
        $newName = $name;
        $newName =~ s/\..*//;
        $newName .= ".xml";
        my $new_fqname = catfile($workdir, $newName);
        $result = rename($fqname, $new_fqname) 
            or die "Cannot rename $fqname to $new_fqame: $!"\n;
    }
}

$|=1;    # Turn off I/O buffering
ScanDirectory($ARGV[0]);
[download]

Better yet, let's remove needless recursion.

use File::Spec::Functions qw( catfile );

# This subroutine takes the name of a directory and recursively scans
# down the filesystem from that point looking for files named "core"
sub ScanDirectory{
    my $workdir = shift;
    if ($workdir =~ /^[a-zA-Z]:\z/) {
       $workdir .= '.';   # Fix bug in Windows.
    }

    my @todo = ( $workdir );

    while (@todo) {
        my $workdir = shift(@todo);
        print "Work dir = $workdir\n";

        opendir(local *DIR, $workdir)
            or die "Unable to open $workdir: $!\n";
        my @names = readdir(DIR)
            or die "Unable to read $workdir: $!\n";
        closedir(DIR);

        foreach my $name (@names){
            next if ($name eq ".");
            next if ($name eq "..");
            my $fqname = catfile($workdir, $name);

            if (-d $fqname){                  # is this a directory?
                push @todo, $fqname;
                next;
            }

            print "Name = $name\n";
            $newName = $name;
            $newName =~ s/\..*//;
            $newName .= ".xml";
            my $new_fqname = catfile($workdir, $newName);
            $result = rename($fqname, $new_fqname) 
                or die "Cannot rename $fqname to $new_fqame: $!"\n;
        }
    }
}

$|=1;    # Turn off I/O buffering
ScanDirectory($ARGV[0]);
[download]

Btw, I properly scoped DIR (added local *DIR;) and removed requests to ignore prototypes (removed &).

Update: Added fix.

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