Here is the code, commented and with follow the progress prints. There are ~35 code lines, the rest are comments, prints or whitespace.
As hinted at in my short poem, the secret (such as there is one) is to analyse the problem.
THREE must be a five digit number, but also it must be a 5 digit number that
ends with the same two digits. With this we remove 90% of the possibles
for the letters T-H-R-E, in fact there are a mere 27 in all as you can see if you run the code.

As we go we just push the possibilities for our letters into an array, we then split the letters out as required.

TEN uses the same T and E as found in T-H-R-E-E so we only
need to search for N within the constraint of Ts and Es found initially. Once we have
our possibles that satisfy T-H-R-E-N we look for ONE. Once again we only have to
look for O within the constraint of the Ns and Es already generated.

At this stage there are only six possible combinations of digits for T-H-R-E-N-O.
We get rid of 4 as they contain the same digit for two or more letters to leave just two.
We then brute force the possibilities for SIX from the remaining 4 digits (only 4*3*2 = 24 cases)
to get the answer.

Tachyon

#!/usr/bin/perl -w
use strict;
my $time = time();
my (%tri,@pos,@pos1,@pos2,@pos3);
# make hash of tiangular numbers 5 digits or less
# the 447th has 6 digits so we don't map past 446
map{$tri{.5*$_*($_+1)}=1}1..446;
# find all possible matches for 'three'
# these are 5 digits long, but last two digits are the same
# this allows us to limit the search
for my $key(keys %tri){
push @pos,$1 if $key =~/(\d\d\d(\d)\2)/;
}
# let's see how many possibilities we have
print "Initially we have ".@pos." possibles for \$t\$h\$r\$e\$e\n";
print "$_\n" for @pos;
# find all possible matches for 'ten' within constraint
# of $t and $e possibilities generated above, we are looking for 'n'
for (@pos) {
my($t,$h,$r,$e)=split'',$_;
for my $n(0..9) {
push @pos1, "$t$h$r$e$n" if defined $tri{"$t$e$n"}
}
}
# let's see how many possibilities we have left
print "\nNext we have ".@pos1." possibles for \$t\$h\$r\$e\$n\n";
print "$_\n" for @pos1;
#now look at 'one' in same way, we are looking for 'o'
for (@pos1) {
my($t,$h,$r,$e,$n)=split'',$_;
for my $o(0..9) {
push @pos2, "$t$h$r$e$n$o" if defined $tri{"$o$n$e"}
}
}
# let's see how many possibilities we have left
print "\nNow we have ".@pos2." possibles for \$t\$h\$r\$e\$n\$o\n";
print "$_\n" for @pos2;
# remove dulicates where digits for $t$h$r$e$n$o are not unique
# I'm sure there is something more elegant but this works
for (@pos2) {
$_ =~ /(.)(.)(.)(.)(.)(.)/;
push @pos3,$_ if $_=~m/[^$2$3$4$5$6][^$1$3$4$5$6][^$1$2$4$5$6][^$1
+$2$3$5$6][^$1$2$3$4$6][^$1$2$3$4$5]/;
}
# let's see how many possibilities we have left
print "\nAfter removing cases where we have duplicate digits\n";
print "we have ".@pos3." possible matches for \$t\$h\$r\$e\$n\$o\n";
print "$_\n" for @pos3;
# find the solution
for my $pos(@pos3) {
# get the remaining digits available for 'six'
# we erase the 6 digits we are currently using
# for t h r e n o
my $remaining = '0123456789';
for (split'',$pos) {$remaining =~ s/$_//;}
# look at the remaining cases
print "\nBrute forcing\n";
print "If \$t\$h\$r\$e\$n\$o\ is $pos then \$s\$i\$x must come fro
+m $remaining\n\n";
# brute force possibilities for six, it's only 4 digits
my @rem = split'',$remaining;
for my $s(@rem){
i: for my $i(@rem){
next i if $i==$s;
x: for my $x(@rem) {
next x if $x==$i or $x==$s;
if (defined $tri{"$s$i$x"}){
my($t,$h,$r,$e,$n,$o)=split'',$pos;
# prove we are right!
print "\nfound solution\n";
print "###################################\n";
print "one $o$n$e " if defined $tri{"$o$n$e"};
print "three $t$h$r$e$e " if defined $tri{"$t$h$r$
+e$e"};
print "six $s$i$x " if defined $tri{"$s$i$x"};
print "ten $t$e$n\n" if defined $tri{"$t$e$n"};
print "###################################\n\n";
} else {print "No match \$s\$i\$x -> $s$i$x\n"}
}
}
}
}
$time = time()-$time;
print "\nElapsed $time seconds\n";

If anyone has an elegant way of deleting the cases where we have duplicate characters I would love to see it.
My 2 line regex is functional, but fairly agricultural!