Parsing Symphony Chat with Perl

One way to pull out what you need is a regex, given that you have clear anchors

my @parts = $string =~ /
    ([0-9]+)             # numbers for day
    (?:[^0-9]+)?\s+      # for st|nd|rd|th (optional!), space. not captured
    (\w+)\s+             # month
    ([0-9]+)\s+          # year
    .+?                  # the rest, but only up to the initials 
    ( (?:WJ|NW|JK|KB|II): .+? INC[0-9]+ )
/x; 

Some patterns here can be strengthened, or weakened (for one, instead of an alternation of expected initials we could have [A-Z]+:, allowing for others and for more letters).

Then convert time to the desired timestamp. A good tool for that is Time::Piece. Altogether

use warnings;
use strict;
use feature 'say';

use Time::Piece;

my $string = q(16th Jun 2020 7:57:18 am Tom Lewin: WJ: RE: BART failed STP - JIMBCI - INC101981467816th Jun 2020 8:20:38 am Nathan Winslow: II : RE: Loans are experiencing issues sending RUNZ - INC101981521816th Jun 2020 8:57:58 am Nathan Winslow: NW : RE: FW: Missing pool factor PnL [Restricted - Internal] - INC101981603016th Jun 2020 9:13:49 am Nathan Winslow: JK : RE: missing sales credits - INC101981633816th Jun 2020 9:24:26 am Nathan Winslow: KB : RE: Bookbuilder not responding - INC1019816567);

my @parts = $string =~ /
    ([0-9]+) (?:[^0-9]+)?\s+ (\w+)\s+ (\w+)\s+ .+? 
    ( (?:WJ|NW|JK|KB|II): .+? INC[0-9]+ )
/x;
#say for @parts;  say '---';

my $dt = Time::Piece->strptime("@parts[0..2]", "%d %b %Y");

say $dt->mdy('/'), ' ', $parts[3];

The last bit is perhaps better done as

my $date = Time::Piece
    -> strptime( join(' ', splice @parts, 0, 3), "%d %b %Y")
    -> mdy('/');

say "$date @parts";

where now we don't have to count off the exact number of elements to print.

In this case @parts in the end has just one element, but changes to requirements do happen. Also, if some elements are in fact needed on hand as separate for other purposes (add sets of capturing parenthesis) then @parts will have more elements.

These print what is needed.

The easiest way would be to "fix" the line before printing it.

So add

use Date::Parse;
sub fix_line {
  my $output = shift;
  if ($output =~ s/^(.*? (am|pm)).*?\s([A-Z]{2} ?:)/$3/) {
     my @date = localtime(str2time($1));
     $output = sprintf("%02d/%02d/%02d %s", @date[4]+1,@date[3],@date[5]-100,$output) ;
  }
  return $output."\n";
}

at the bottom of our string and change your inner fer-loop to:

my $nl = '';
foreach my $word(@words) {
   if($word =~ /\d{2}(?:st|nd|rd|th) \w{3} \d{4}/a) {
        chomp $word ;
        $nl.= $word ;
    }
    else {  
         print fix_line("$nl$word");
         $nl=''
    }
}