Define 'fast' and/or 'timely manner'. I guarantee the following will complete before lunch time:

use strict;
use warnings;

my @dirs;

while (my $line = <DATA>) {
    my ($indent, $entry, $isDir) = $line =~ m!^(\s*)([^/]*)(/|\\)?!;

    next unless defined $entry and length $entry;

    my $depth = length ($indent);

    if ($depth < @dirs) {
        splice @dirs, $depth, $#dirs;
    }

    if ($isDir) {
        push @dirs, $entry;
        next;
    }

    print join ('/', @dirs, $entry);
}


__DATA__
base/
 dir1/
  file1.txt
  a.png
  dir2/
   b.txt
  q.txt
 dir3/
  a.png
 f.txt
[download]

Prints:

base/dir1/file1.txt
base/dir1/a.png
base/dir1/dir2/b.txt
base/dir1/q.txt
base/dir3/a.png
base/f.txt
[download]

Update: using jdporter's assign to $#array technique the code reduces to:

use strict;
use warnings;

my @dirs;

while (my $line = <DATA>) {
    my ($indent, $entry, $isDir) = $line =~ m!^(\s*)([^/]*)(/|\\)?!;

    next unless defined $entry and length $entry;

    $#dirs = length $indent;
    $dirs[-1] = $entry;
    next if $isDir;

    print join '/', @dirs;
}
[download]