Why does the user not get a "transaction complete" or "success" confirmation message after submitting an online transaction?
The answer is that I'm not sure why this is no longer working for you. 


All our 4.6 sites have been upgraded to 4.6.26 and are using iATS
1.5.3 and I've had no reports of anything breaking. 
I've just confirmed myself by making a donation to one of our projects that I
get to the Thank you page.


There is a bit of Javascript on that Make Contribution button. Ask your developer/support to open up a Firebug/Console to start the hunt as to what may be going on with your Contribution page.

From your Comments I see DOMPDF is throwing an error -> it is known to blow up at times - best to disable PDF creation for your receipt. It's one of those obscure settings under Misc -> civicrm/admin/setting/misc?reset=1 -> hit No on Attach PDF copy to receipts.